L5E
E15 and E35 are methods that solves the last six or the last eight edges of the U and D layers, first three D layer edges are done (one, BD in E15) and then the last five ones. Always preserving all corners and Eslice edges. The first part also solves the centres.
E35 is not a compleate brute force method, it uses a combination of intuition and algorithms, an advanced user even include ELL in the method to save turns in some cases but most times you end in EPLL or 5 edge permutation. It is much as Roux last step but here orientation is done after all centres are solved to make recognition easier/faster so the Mpermut used last in Roux is not really useful here (you solve centres and BD before orient instead of RU/LU after).
Short description:
 E3 part First pair up RD and LD edges, then place them together with R and L centres or simply do one at the time, the first one as a pair with the centre and the second using AUF M' U/U'/U2 M, last palce BD + B centre as a pair.
 E1 part As E3 but only the last edge.
 E5 part this part has got two main subparts, first orient the edges using the Roux M' U M style (5 cases, newer the worst 6 edge case of Roux). If possible also place FD during orientation, else do it after using M' U2 M and end in EPLL. You can also first orient and then permute all five edges in one go.
E35 may be used in CF if the Eslice is solved after the corners or in any method that solves columns first, like doing all pairs first and then CMLL (or CMSLL). E15 is a good alternative to Roux last step. The E5 part is a stand alone method, it can be used togehtr with a standard F2L where the last edge for the cross is ignored and LL corners are solved after that (CLL or CMLL).
Contents
Intermediate system for E5
Intuitive:
This intuitive style is not all intuitive because you need to know EPLL. First orient all unorientd edges, then place the FD edge to it's correct position and end the solve in EPLL. For that you basicly need this:
 P = M' U2 M
 O = M' U M
 O' = M' U' M
 O2 = P ... not used
From that plus some U turns you can solve any case, (including all ELL cases), example: O U P = (M' U M) U (M' U2 M). O solves orientation U P places FD edge and then the EPLL solves permutation for the last four edges, all now in LL (if it did not skip, it is a 1:12 chance).
The algorithms for some cases can be shorter if you also add this to the system:
 F() = F2 ( MOVES ) [u] F2
The U turn can be any U, U' U2 or no U turn depending on how U moves inside the parentesis of the "function". F(U P) makes the usual 7 turn UPLL = F2 (U M' U2 M) U F2. F(O) is a nice way to solve orientation and place FD in one go if it is in the middle of three unoriented edges in LL, just AUF as you where about to do only O.
Using algorithms:
The intuitive style lets you find and understand the algorithms you need to orient the 5 cases that are possible for the orientation part but another approach is to learn the algorithms instead of finding them.
Sometimes you will solve FD permutation while doing this, in the images the yellow sticker shows where the DF edge has to be to get solved using the given algorithms (for all other reasons, treat it as a white sticker). If you learn these position you can for some situations also learn to use the mirror algorithm when the FD edge is on the opposite side from where you normally solve it, the easiest example is to use O' (M' U' M) instead of O (M' U M). In the cases where you cannot solve FD permutation while orienting you use AUF + P to solve it, it is at the most 4 turns, then EPLL.
Semi advanced system for E5
Orientations with FD permutation:
In many cases you can solve FD permutation wile orienting in the same number of turns as you do only orientation. For the rest of the cases it is a bad idéa to learn algorithms because they would most times end in the same turns as do anyway to solve FD after orientation or if not the length of the algorithm is of the same length as the 2step or at least almost. But there are a number of good cases, here is the list:
 alg ... (possibly optimised) MOVES ... orient pices, permute FD
 M U F(U' M') ... M d R2 d' M' d R2 ... RU and FU, FD at UF.
 M U' F(U M') ... M d' L2 d M' d' L2 ... LU and FU, FD at UF.
More algs later...
EP5
EPLL + FD permutation in one go:
Another reason for not learning all algorithms for placing FD while orienting is that you can do the permutation for all last five edges in one go using only 9 extra algorithms above the usual 4 EPLL's, a total of 13 algs, add to that the needed 5 orientations and you got a system of 18 algorithms to solves E5 in two short steps. But it is still recomended to use as many of the short algorithms for orientation + FD permutation as possible because recognition is faster for EPLL than it is for most the of rest of the cases.
Here follows the algorithms for permutation of all 5 edges, preceed these with AUF to place the FD edge at the position it has in the cycle decription after the alg.:
But first we also introduce this:
 P = M U2 M'
And this:
 f() = f2 ( MOVES ) [U] f2 ... the same as the F() function above but double layer f turns.

 

 

 
 F(ZPLL) ... (y') r2 U 2x(M E2) U' r2 ...
 F(Z'PLL) ... ...
Three cycles:
 P ... M' U2 M ... UF>DF>UB>UF
 f(d P) ... (y') r2 U M' U2 M U r2 ... UR>UB>DFUR
 f(d' P) ... (y') r2 U' M U2 M' U' r2 ... UL>UBDF>UL
Five cycles:
 P U P ... (M' U2 M) U (M' U2 M) ... UF>UL>URDF>UBUF
 P U' P ... (M' U2 M) U' (M' U2 M) ... UFURULDFUBUF
More algs later...
Advanced system for E5
The advanced style is to create algorithms for all extra ELL cases that solves E5 in one go. This has not ben done by anybody yet, it's your chance to get famous, create the system and post the algorithms with descriptions here =)
Don't think about it, it is so many cases and for the worst ones you will end up in an alg having exaclty the same turns as you do 2step anyway. Also, the one step approach has got much slower recognition, the benefit of trying to learn is wery small, just a lot of work.