On the 4x4x4 and larger puzzles, single-layer turns are not enough to fully scramble or solve the puzzle. Thus there are notations to describe turns of multiple layers on the same face. A double-layer turn, for instance, might include both the outside layer and the parallel layer touching it.
Double-layer turns are also popular in algorithms for the 3x3x3. There are two main ways to write this, using the U face as an example:
Uw - this notation is widely used by Japanese cubers and was recently adopted by the WCA.
u - this notation is often preferred by English-speaking cubers, and originally appeared for writing down 4x4x4 and 5x5x5 moves.
For turns of more than two layers, it is impossible to use single letters to write down turns, but a popular proposal is to use numbers to show the number of layers that are being turned. There are several ways to write these turns, using the U face as an example and 3 as an example number of layers, but the best is either 3U or 3u, because putting the number after the letter can cause confusion when there is a half turn (giving it the suffix 2).