We know that this thread may induce the “wall of text” syndrome (try opening up all the spoilers), but we promise that you are doing yourself a favor by reading this. There is a lot of information, but it is all beneficial and there is no unnecessary information. This thread is intended for those just starting out, but it may also be useful to those looking to switch methods, or those who just want to learn about other methods.
Introduction to Thread
There are advantages and disadvantages to each method. Take a look through, and identify the combination that appeals to you most.
“Fast times” can be set with any method. Remember the very first World Record was set with a peculiar Corners First variant, and at the time it was considered unbreakable. Today the World Record is set with CFOP, and it is also seems unreachable to most. This pattern may continue. Jessica Fridrich herself thought the limit to her method was a 13 second average, so only time will tell how speedcubing will evolve. As the great Alexander Lau once said: “Methods don’t have speeds.”
Don’t base your selection on method “popularity.” Just because everyone at your school uses the Roux method, or all your favorite YouTube cubers use the CFOP method, or your two best friends use the ZZ method, that doesn’t mean that the Petrus Method isn’t compatible with you at all. Cubing is about fun and personal fulfillment, so their decisions ought not to affect you whatsoever.
Do external research! As much as we have tried to be clear and comprehensive in our explanations, there may be tiny details that we have overlooked. If you have any questions or confusions about the content of this post, we urge you to check the Speedsolving Wiki and search the Speedsolving Forum for more information.
Please skim these abbreviations to ensure you will extract as much knowledge from this reference as possible. These abbreviations are standard in cuber communication, so a quick skim will not be in vain.
“EO” abbreviates “Edge Orientation,” in reference to the direction of a defined sticker on an edge piece.
“CP” abbreviates “Corner Permutation,” in reference to the relative position of corner pieces.
“LL” abbreviates “Last Layer,” in reference to the Last Layer to be solved on a cube.
“Perm” abbreviates “Permutation Algorithm,” in reference to an algorithm used to solve the last step of the CFOP/ZZ/Petrus methods.
“Alg” abbreviates “Algorithm,” in reference to a move sequence applied to a cube.
“The Morse Code” is an anagram for “Here Come Dots.” Fishy, eh?
The Cross includes placing 4 edgepieces containing a single color onto that side of the cube, such that the stickers on both sides of the edge piece are “touching” the center piece of the same color. This cross is ultimately placed on either the bottom or left side of the cube, depending on where the solver would like to complete the next stage of the puzzle. 

The next stage is the F2L. The solver pairs corner and edge pieces that belong in the F2L “slots” that they created during the previous step. These pairs can be built and inserted intuitively or algorithmically. The solver usually must complete 4 F2L pairs, but by using advanced techniques, this step can be greatly shortened. 

The next step is to Orient all the pieces that reside on the Last Layer. There is an set of 57 OLL algorithms that provides for the completion of this step with 1 move sequence. For beginners, however, there is a 2step approach that involves only 9 algorithms. 

The final step is to Permute all the pieces that reside on the Last Layer. There is a set of 21 PLL algorithms that provide for the completion of this step with 1 move sequence. For beginners, however, there is a 2 step approach that involves only 6 algorithms. 

Algorithms can be learnt for all steps after the cross, minimizing the amount of thought required.
F2L can be learnt intuitively or algorithmically depending on preference. In the end, you will store this information in muscle memory, so the learning method you choose is irrelevant.
Large amount of resources (both video and textstyle).
Easy extension of the LayerByLayer beginner’s method.
Disadvantages:
 Lots of algorithm to learn
 Not very move efficient
 Tends to have several cube rotations (primarily yrotations, the slowest rotation) per solve.
“I use CFOP for 3x3 solving because it's the first method I learned, and I've practiced very much with it, and I think at this point there is no point in switching methods unless there is a major breakthrough ultra amazing method.”  Antoine Cantin
“For 2 Handed solving, I think that something like roux would probably be faster than CFOP since there are almost no rotations and your moves are very quickly reduced to (R, U, r, M, F) which can be very fast to execute. Also, the move count is generally lower.”  Antoine Cantin
“Don't give up understanding and optimizing your F2L (or any other part of the solve) by trying out new stuff or doing research. Find out your weaknesses and try to improve them the most. PRACTICE”  Antoine Cantin
You are good at learning algorithms
You are the kind of person who would rather use the most popular method
http://badmephisto.com  Badmephisto’s websites, with links to many helpful tutorials for beginners for the intuitive parts of CFOP as well as OLL and PLL algorithms. However, some content is a bit outdated, such as the recommended cubes.
http://cubewhiz.com  A great resource that contains much helpful information including OLL and PLL algortihms and recognition.
http://cubefreak.net/speed/cfop/  An overview of CFOP that contains printable OLL and PLL algorithm sheets.
OLL
http://www.youtube.com/watch?v=DTYvklyOpVM  A guide to 2look OLL, which you should learn before then going on to learn full OLL
http://www.speedsolving.com/wiki/index.php/OLL
PLL
http://www.youtube.com/watch?v=S61q3FYVFis  A guide to 2look PLL, which you should learn before then going on to learn full PLL
http://www.speedsolving.com/wiki/index.php/PLL  These two links contain multiple algorithms for every OLL and PLL case, so that you can choose which one you like the best for each case.
http://www.youtube.com/watch?v=qBYycb7hR4Y  How to recognize the PLLs.
F2L
http://www.youtube.com/watch?v=kxbcAMfWwM  Part 1 of a very good tutorial on intuitive F2L.
http://www.speedsolving.com/wiki/index.php/F2L  This link contains multiple algorithms for each F2L case, in case you would like to learn algorithmic F2L.
http://cubefreak.net/speed/advancedf2l/  Great guide to advanced F2L.
In CFOP, color neutrality means that you can do a cross on any side. Dualcolor neutrality refers to being able to solve a cross on either of two opposite sides. Fixed cross means that you can only solve the cross on one side, usually white. According to Lars Vandenbergh’s analysis of color neutrality in CFOP, here are the average optimal movecounts to solve the cross for each level of color neutrality:
Average number of moves to solve fixed cross: 5.81
Average number of moves to solve dual cross: 5.39
Average number of moves to solve neutral cross: 4.81
So, on average, being color neutral takes one move off of your cross. This may seem trivial, but according to multiple color neutral sources such as the 3x3 world record average holder, Feliks Zemdegs, it helps the whole solve flow better because you can choose the best cross, leading to a better CrossF2L transition.
Almost everyone agrees that when you are starting out, it is best to start out color neutral because of the slight advantage it will give you in the long run. Once you get very fast with fixed cross, it is very difficult to switch to neutral cross. Dual cross is a good compromise between the two, but it is recommended that if you are reading this and are just starting out and choose CFOP, you should be color neutral.
Cross on Left  With this variant the user builds the cross on the left face, as opposed to the down face in standard CFOP. This allows the F2L to be solved using primarily <R, U, x> moves. Rotations on the xaxis (i.e., xrotations) are generally faster than the yrotations that would be used otherwise, however, when the F2L is finished, the solver must perform a z' to place the last layer on the top of the cube for OLL and PLL.
CFCE  Instead of solving the last layer using OLL and then PLL, you solve the corners using CLL and then the edges using ELL. The reason that this isn’t used very often is because of slower recognition and algorithms that are harder to fingertrick. However, some very experienced CFOP users know ELL just in case the corners of the last layer are solved, so they can 1look LL.
CLL algs: http://www.speedsolving.com/wiki/index.php/CLL_algorithms_(3x3x3)
ELL algs: http://www.speedsolving.com/wiki/index.php/ELL
VHLS  Involves learning algorithms to orient edges while inserting the last F2L pair. This leaves you with a “top cross,” similar to what you get when you finish F2L with ZZ or Petrus. Involves learning 32 algorithms.
Algs: http://www.cubewhiz.com/vh.php
ZB  Involves learning several algorithms for every last slot F2L case to force all edges to be oriented  this part totals 125 algorithms alone, not including mirrors and inverses.. Full ZB then involves learning an algorithm for every possible case on the top that could result  this is called ZBLL, and can be combined with ZZ or Petrus. ZBLL contains almost 500 cases. Very few people have ever learned ZB, and no one has successfully learned it and gotten relatively fast at it. “A fate worse than death.”
ZBLS: http://jmbaum.110mb.com/zbf2l.htm
ZBLL: http://jmbaum.110mb.com/zbll.htm
MGLSF : While inserting the final F2L pair, you solve the OLL, leaving you with just a PLL. The two steps are ELS and CLS. You insert the F2L edge while orienting the edges(ELS), then you insert the F2L corner while orienting the corners(CLS). This just leaves you with a PLL. This method, as far as we know, has never been put into practice. This method requires memorization of 69 algs, not including mirrors.
MGLS: http://cube.garron.us/MGLS/
The First Block consists of building a 1x2x3 block all sharing a single color. The block is usually placed on the left side of the cube, though some practicioners place it on the right (usually a difference between right and left handed solvers). 

The next step is the Second Block, consisting of another 1x2x3 block sharing the opposite color of the first block, placed on the opposite side of the cube (usually the right). This step must be performed using only R, r, M, and U moves so as to preserve the first block. 

The third step is CMLL. This step solves the corners of the last layer, disregarding the Mslice and Ulayer edges. Though there are 42 algorithms, because of these freedoms, the algorithms are often shorter and faster than standard COLL. For beginners, this step is often broken into two parts, orienting and permuting separately, using only 9 algorithms. 

The final step is to solve the Last Six Edges (LSE). This step is done in 3 substeps: 1) Orienting the edges so they can be solved using M2 and U moves or M and U2 moves; 2) solving the UL and UR edges; 3) solving the last 4 edges and centers in a single, intuitive algorithm.  
The edges are oriented when only U and D colors are on the U and D faces of the cube. This is probably the most difficult step of CMLL to perform intuitively. 

Solving the UL and UR edges is a simple step, but must be performed while preserving the orientation of the edges. After this step, the left and right sides of the cube will be completely solved. 

The final substep is to solve the remaining edges and finish the cube. This step is very fast and can usually be performed in 4 moves. 

 Very move efficient
 Few Algorithms
 No rotations
 Very intuitive
Disadvantages:
 Not as algorithmic so good lookahead is required to be fast
 Requires lots of thinking at the when first starting because blockbuilding is involved
"In Roux we have bow ties. Roux  1, CFOP  0" PandaCuber
“[When you first start,] take it slowly and make sure your movecount is good enough. Speed comes naturally.”  Alexander Lau
“Opposite colour neutral (8 first blocks) is a good compromise for me simply because it balances inspection length with freedom of choice.”  Alexander Lau
You are not good at learning algorithms
You want to solve the cube with a mostly intuitive method
You want to finish your solve with some sexy M slices
http://wafflelikescubes.webs.com/  A great site filled with lots of information about Roux, containing a written guide to Roux, CMLL algorithms (2look and 1look), and a guide to sub15
http://grrroux.free.fr/method/Intro.html  Gilles Roux’ personal site, giving his own explanation and tutorial for the method.
http://www.speedsolving.com/wiki/index.php/Roux_Method  A good webpage giving some pros, cons, and simple variations of the method
http://www.speedsolving.com/forum/showthread.php?11506WafflesRouxTutorial  Waffle’s video tutorials for the Roux Method
http://rouxtorial.webs.com/  5BLD (Alexander Lau) and PandaCuber’s (Bryan Rusinque) Roux tutorial. It contains a full, in depth guide to the entire Roux method.
http://www.speedsolving.com/forum/showthread.php?20355CMLL  A SpeedSolving thread containing many CMLL algorithms for each case to help find the best for each cuber.
https://www.youtube.com/watch?v=mBy0XQiN0M&list=PLajHGvYF36nSsL1r_DqrpDY07TnJwqEpn  DeeDubb's Roux tutorial. A guide to solving using the Roux method aimed at complete beginners
Very few people are fully Color Neutral with roux due to the difficulty during the blockbuilding step. There are two common forms of partial color neutrality in Roux. The first is <x2, y, z2> neutrality, which is similar to opposite neutrality in CFOP  the solver uses only opposite colors for the U and D faces (e.g., using white/yellow for U/D). The other form of partial color neutrality is <x, y2, z2>, or using only opposite colors for L and R faces. Though each form gives a mathematically equal number of different options for the blocks, it is generally agreed that <x, y2, z2> gives more options for the blocks, though makes CMLL recognition more challenging.
CMLL+EO is an extension on CMLL that orients the edges during CMLL so that step can be skipped during LSE. However, the large number of algorithms required for each CMLL case cause few people to decide to learn this. It is comparable to learning ZB or OLLCP in the CFOP method. http://www.speedsolving.com/forum/showthread.php?21210KCLL
Nonmatching Blocks is a selfexplanatory name. The second block does not need to match the first block, giving 4 options for the second block, instead of just 1. This allows the second block to be much faster, but makes CMLL, LSE and EO much more difficult. Here is a page about nonmatching CMLL that will occur when using nonmatching blocks: http://www.speedsolving.com/wiki/index.php/NMCLL
The EOLine is the most distinctive part of the ZZ method. In this step, the solver orients all the edges while placing the DF and DB edges. The two edges and the bottom centre are the "line" in EOLine. This step puts the cube into an <L, U, R> group, meaning F, B, or D moves are not required for the remainder of the solve. Although this step may seem like a hinderance, it speeds up the F2L and LL. 

The next stage is the F2L. The solver creates a 2x3x1 block on each side of the line via blockbuilding. Because one only needs to do L, U, and R moves, solving can be very quick. 

The next step is to Orient all the corners that reside on the Last Layer. There are only 7 algorithms for this step because the edges are already oriented. Some solvers choose to learn COLL, which solves the corners while keeping the edges oriented, leaiving a case that only requires the edges to be permuted. 

The final step is to Permute all the pieces that reside on the Last Layer. There is a set of 21 PLL algorithms that provide for the completion of this step with 1 move sequence. For beginners, however, there is a 2 step approach that involves only 6 algorithms. 

No rotations
<R, U, L> F2L  Meaning you only have to turn Right, Left, and Up faces during F2L  this can be very good for One Handed solving
More last layer solution possibilities because of solved edge orientation
Disadvantages:
EOLine (first step) is harder than cross
Transition between R and L faces can sometimes be awkward
More pieces to solve after EOLine than after cross
“I think ZZ is very strong OH (and possibly feet). It doesn't stand out very much in regular 3x3. Big cubes are all very awkward.”Phil Yu
“Study a lot of EO Line solutions and play with an optimal solver. Make good first block choices by studying your options thoroughly. Don't rush.”  Phil Yu
“Color neutrality is kind of limited on ZZ. I use two different fronts[a.k.a. Yaxis] and that requires a lot of thinking and inspection time already. ZZ is very inspectionheavy.”  Phil Yu
You like the idea of an <R,U,L> F2L. This means the cube can be solved with R, U, and L moves without cube rotations.
You are interested in One Handed solving
N.B.: EOLine will feel very hard at first; if you like the method in general, stick with it, and don’t give up  over time it will become very easy. Trust us on this one.
http://cube.crider.co.uk/zz.php : Very good ZZ text tutorial by Conrad Rider.
https://www.youtube.com/playlist?list=PLD9771CF83F13B110  Very good ZZ video tutorial by Phil Yu.
https://www.youtube.com/playlist?list=PLXoQgT40ztta9c6Ol9FRVvxCPQehiQNcv  A collection of videos on various ZZ topics by Phil Yu.
https://www.youtube.com/watch?v=d0vVDBi3_EU  ZZ method walkthrough solves by Phil Yu.
https://www.youtube.com/watch?v=8DaC4Bs2sU  ZZ method walkthrough solves (OH version) by Phil Yu.
http://www.stachu.net/cubing/zzTimer  A timer by Stachu Korick which allows you to generate scrambles with “n” misoriented edges.
http://laire.fi/jarcs/  Can be used to find optimal EOLines and study them. This activity is recommended but not necessary, and should improve your EOLine building skills.
Because recognition for misoriented edges is so dependent on the solver’s orientation, it is very hard to be even partially color neutral with ZZ. Some argue that since a ZZ solve involves no rotations, even if one were to become partially/fully color neutral, the advantages may be cancelled out by the fact that one gets very familiar with their orientation with ZZ, and subconsciously knows where each piece belongs. However, here are some more realistic types of color neutrality than full color neutrality:
<x2, y2, z2>:
This means that you can do an x2, y2, or z2 from your starting orientation before starting EOLine. This allows the recognition for bad edges to stay the same no matter what orientation you solve in, so you would choose which orientation to solve in based on the convenience of line edges. While this is arguably the easiest of the color neutrality variants listed here, it doesn’t present much of an advantage because the line edges take so few moves to place.
Y axis:
This means that you can do a y (or y’) from your starting orientation before starting EOLine. This only changes the recognition of orientation of the E layer edges, and allows you to solve F2L with the same side on the bottom every time. Phil Yu is working on switching to full Yaxis color neutrality.
Z axis:
This means that you can do a z (or z’) from your starting orientation before starting EOLine. Because you keep the same faces on F and B, you do not have to recheck for orientation of edges, the same edges will be misoriented, so you don’t have to recheck for orientation. This allows you to choose the best EOLine on the Z axis, although for each EOLine, you will have the same amount of misoriented edges to deal with.
Overall, it is agreed that color neutrality is not recommended for ZZ, although Y axis is probably the most practical and advantageous of the options listed above.
Because the ZZ method reduces the cube so efficiently and effectively to a restricted movegroup, it is a springboard for a host of legitimate variations towards solving the last F2L block, the last layer, and even a combination of the two.
ZZWV or ZZWinter Variation is a variant that involves forming the last F2L pair, and inserting it with a special algorithm that will also orient the last layer corners. In effect, this variant gives you an OLL skip every time. There are 27 different algorithms that need to be applied instead of the standard R U’ R’, but because your last F2L pair can require insertion in any of four different slots, you will have to be able to mirror these 27 algs and also be able to perform them from the back. Most experienced cubers know at least the easiest Winter Variation cases. Some cubers also know the Summer Variation which performs the same function as Winter Variation, but for the R U R’ pair insert instead of R U’ R’. See the Winter Variation article on the Speedsolving Wiki for more information.
ZZb is the Last Layer approach that Zbigniew Zborowski originally intended for ZZ. It involves phasing the Last Layer edges during the insertion of the last F2L block. This means that opposite Last Layer edges will be opposite from one another. That reduces the possible Last Layer cases by a factor of three. This is a pretty good deal, considering the user only invested a few extra moves. The algorithm set ZZLL, including a whopping 160 distinct cases, is then applied to solve the cube. Though this may seem like a lot, half of the algs can be discounted because they are mirrors of other algs. And half of the remaining algs can be discounted because they are inverses of other algs. In the end, this variant has great potential, but few cubers have ever travelled this route and lived to tell the tale. See this forum post for more information..
ZZa is a fate worse than death. Through brute force and rote algorithm memorization, the solver applies one of 493 different algorithms to solve the Last Layer in one move sequence. There’s no smoke and mirrors, no setup moves, no presteps, nothing. You just finish F2L, recognize the LL case, and apply the final algorithm. If you choose this variant, prepare to never smile, laugh, or breath fresh air ever again. You have been warned. See the ZBLL article on the Speedsolving Wiki for more information.
ZZCOLL/EPLL is the Last Layer approach that most ZZ speedcubers use. It is a decent compromise between algorithm count and algorithm speed. COLL abbreviates Corners Of Last Layer, and it is used to solve the Last Layer corner orientation and permutation at once. There are 42 algorithms. EPLL abbreviates Edge Permutation of the Last Layer, and it used to place the Last Layer edges in their correct positions. There are 4 algorithms for this step, all of which can be performed 2generator (meaning only 2 faces of the cube need be turned to solve the cube from this state), making this step extremely ergonomic and quick, especially for OH (One Handed) solving. However, because the traditional Sune and Antisune OLL algorithms are so short and easy to execute, many experienced speedcubers forgo memorization of the Sune/Antisune COLL algorithms. See the COLL and the EPLL articles on the Speedsolving Wiki for more information.
ZZd is a very complicated method. The idea is that cubers love <R,U> moves more than rollercoasters, chocolate, and girls. So, why not place the cube in a state than can be solvable with <R,U> moves as soon as possible during a solve? There are many subvariants that incorporate different algorithms and approaches to solve the corner permutation (the characteristic of the cube that defines the state of 2generator solutions). However, the two most notable were proposed by Sebastiano Tronto, who goes by Porkynator on the Speedsolving Forums. ZZPorky v1 involves solving the corner permutation after completion of the first F2L block. There are 2 algorithms and a pretty easy recognition system. However, this approach requires many moves. ZZPorky v2 involves solving the corner permutations during completion of the first F2L block (as was the original point of ZZd, to make the solve 2generator as soon as possible). This approach is much more efficient, but requires knowledge of around thirty short algorithms (3 <R,U> moves maximum each). ZZd is still undergoing very much development. Maybe you’d like to help out!
ZZOCELL/CPLL is a variant so unpopular that you’d bet it was a wellkept secret. However, you’d be wrong. This published variant involves solving the Last Layer in a seemingly opposite order to the standard COLL/EPLL. OCELL abbreviates Orient Corners (and permute) Edges of the Last Layer. All of the 40 algorithms for this step can be 2generator. CPLL abbreviates Corner Permutation of the Last Layer. There are 4 algorithms for this step, including both A permutations, an E perm, and an H perm. See this external resource for more information.
ZZBlah is a variant with a somewhat misleading intent, proposed by Chester Lian. The idea is for the solver to purposely misorient corner pieces on the Last Layer while solving the last F2L pair. Forcing all Last Layer corners to be misoriented reduces the quantity of possible Last Layer states to strictly the H and Pi OLL cases (just 2 of the 7 OCLL cases). The solver then applies the corresponding ZBLL (one look Last Layer algorithm) to solve the cube in one move sequence. There are 112 algorithms for this variant, but half of these may be discounted because they are simply mirrors of other algorithms. See the SpeedSolving thread for more information.
The 2x2x2 Block is the first step of the Petrus method. It is completely intuitive, although there are some tricks that you will learn over time. The 2x2x2 block can be built anywhere on the cube, and therefore the optimal 2x2x2 block is often selected during inspection time. 

The next stage involves expanding the 2x2x2 block to a 3x2x2 Block. The additional 1x2x2 solved portion can be added to any of the three different sides of the 2x2x2. This makes solving the 3x2x2 an extremely versatile step. Advanced Petrus users can also plan this step during inspection time. 

The next step is to Orient the Edges. This is done by placing the solved 3x2x2 block on the back of the cube and using (RUR’) triggers to correct “bad edges.” Advanced Petrus users can solve some of the Edge Orientation during the 3x2x2 block and often take advantage of move cancellations during this stage. 

The next step is to finish solving the F2L. This is done by placing the solved 3x2x2 block on the left side of the cube and using <R, U> moves to quickly and ergonomically complete the F2L. Advanced Petrus users can positively influence this stage during the Edge Orientation step by preserving and creating F2L pairs. Afte completing this step, you will end up with all of the last layer edges in an oriented state, which simplifies the Last Layer solution to come. 

The final step(s) is to solve the Last Layer. Petrus’ original approach for this substep was to solve the corner permutation, followed by the corner orientation, followed by the edge permutation. However, the majority of modern cubers elect to solve the LL with either OLL & PLL or COLL & EPLL. Both are viable options, and the corresponding resources are referenced in the CFOP and ZZ sections of this thread as well as in the Speedsolving wiki. 

More efficient than CFOP
 This method is the general approach to FMC (Fewest Moves Competition)
 Less Last Layer cases exist because edges are oriented during the F2L.
 Very intuitive
 Good foundation for expansion into Heise
Disadvantages:
 Not as algorithmic so good lookahead is required to be fast
 Requires lots of thinking at the when first starting because blockbuilding is involved
 Blockbuilding makes it very hard to be finger tricky
 Not as proven as other 3 methods  no one has gotten sub10 averages using Petrus
“I liked how petrus method relied on efficiency rather than physical speed to get good times.” Erik Johnson
“I think petrus is the best middle ground method between speedsolving and FM solving. it's not the best at either but it works pretty well for both.”  Erik Johnson
“Enter in the weekly FMC and do petrus solutions. I believe that one of the reasons I was able to get really fast with petrus was because I spent several summers doing FMCs. I just spent days and nights on one solution trying different blocks and techniques. This taught me many ways that blocks can be assembled and the repetition taught me to recognize them quickly. Also, don't use the last layer method described on the petrus solving website. it's geared more towards FM solving than speed. I recommend using OLL and PLL.”  Erik Johnson
“Color neutrality is a very important skill for any method. I was only Y/W CN for petrus which allowed me to start with 8 possible 2x2x2 blocks but then limited me to only being able to expand in 2 of the 3 possible directions to make the 2x2x3”  Erik Johnson
You enjoy trying to be efficient
You really like blockbuilding
You want to quickly learn about the way pieces move about the cube.
http://lar5.com/cube/  Lars Petrus’s original tutorial(text).
http://www.speedsolving.com/forum/showthread.php?16141PetrusHowtoFixBadEdges(VideoampText)  How to fix bad edges in Petrus, video and text.
http://www.speedsolving.com/forum/showthread.php?41694NEWPetrusTutorials  New Petrus tutorials by Erik Johnson.
https://www.youtube.com/watch?v=8WCMFwt2khQ  Another tutorial, most popular on Youtube.
Color neutrality in Petrus is often considered essential. However, most of the fastest solvers are only dualcolorneutral. This means that during inspection, they select to start on one of two opposite colors. This still allows them to choose from any of the eight possible starting blocks, and allows them to extend the starting block in 2 of 3 possible ways. Therefore, dual color neutrality, while also being realistic, certainly provides enough of the “neutrality” aspect, just one extending block short of full color neutrality.
Another choice is only starting with a block on one color. This only allows four of the eight possible starting blocks, but allows for 2 of the 3 possible extensions, just like dual color neutrality. The advantages of this almost certainly do not outweigh the disadvantages, so we would suggest dual color neutrality. However, this option is not that bad if for whatever reason you would like to choose it.
JTLE:
After solving F2L without the DR edge, you insert the DR edge while orienting the last layer corners, leaving PLL. This variant has not been explored much at all. The alg count is 27.
Wiki page: http://www.speedsolving.com/wiki/index.php/JTLE
EJF2L:
During F2L, you allow any one F2L corner to be twisted in place. When you finish F2L, you fix the corner while solving orientation of the last layer, leaving PLL. The algorithm count is 24.
Wiki page: http://www.speedsolving.com/wiki/index.php/EJLS
MGLSP:
You insert just the edge of the last F2L pair instead of the edge and the corner, then finish F2L while solving orientation of the last layer corners, leaving PLL. Full CLS requires 104 algorithms, but it is worth it to learn some of the “easy cases” to learn during solves.
Wiki page: http://www.speedsolving.com/wiki/index.php/MGLS
See ZZ Variants for all others.*
*ZZd doesn’t apply to Petrus.
“I don't think method neutrality should be considered for 3x3 speedsolving. each method takes a long much time to master and I don't think you can gain much from choosing an easy cross over a hard 2x2x2. [C]olor neutrality almost eliminates the need for method neutrality because you can just choose the easiest part of the cube to work with. “  Erik Johnson
“It's a silly idea that requires way too much practice and you don't gain anything worthwhile.”  Alexander Lau
“I think that for someone that practices very much, this can be a big advantage since it allows you to have a very good start, depending on the case. However, you'd need to be good and practice very much every method.”  Antoine Cantin
Antoine Cantin(Username=antoineccantin,CFOP, ~10 avg)
Erik Johnson(Username=ErikJ,[formerly]Petrus, ~12 avg)
Alexander Lau(Username=5BLD,Roux, ~7 avg)
Phil Yu(Username=asmallkitten,ZZ, ~13 Onehanded avg)
The following people contributed to the writing and editing of this post:
Matt DiPalma(Username=mDiPalma, ZZ, ~12 avg)
Eli Lifland(Username=uvafan, ZZ, ~12 avg)
Alex Mertz(Username=ickathu, Roux, ~17 avg)
The following person contributed to the editing of this post:
Kevin Costello(Username=KCuber, CFOP ~9 avg)