6x6x6, 7x7x7 ... NxNxN Rubik's cube
The text on this page has been created from some sort of historical reasons. That is, the tutorials for the 4x4x4 and 5x5x5 Rubik's cube have been written at first. Any method, through which we can solve a 4x4x4 cube, may be used (after making certain adjustments) for a solving of even bigger cubes (in terms of a greater number of layers).
That said, I could make a tutorial on how to solve the 6x6x6 Rubik's cube, 7x7x7 Rubik's cube to NxNxN Rubik's cube with the help of reduction solving technique (as in the case of a 4x4x4 and 5x5x5 cube within these pages). But I said no!
Two things brought me to that conclusion - first, it would be enormously time-consuming (I can not imagine that I would make a tutorial for e.g. the 20x20x20 Rubik's cube using simulators) and second, it would be quite useless - a method principle remains the same and only certain layers are being changed when executing moves.
From a personal experience I know what it's like when you mess up a so-called OLL parity algorithm on a 20x20x20 cube. The result is a scrambled cube and eyes full of tears. Therefore, instead of reduction solving technique, I will recommend the Cage method for a solving of "bigger" cubes. It avoids parity problems as known from e.g. a 4x4x4 Rubik's cube.
On that linked page there is explained only a solving procedure for a 4x4x4 cube, nevertheless, it can be adapted to any other NxNxN cube. It is practical to somehow optimize the method on a bigger cubes, for example by solving of pieces of last 4 composite edges (i.e. the edges from a 3x3x3 cube point of view) in multiple layers at once, or by making some kind of "cleaning process" leading to faster solving of center pieces in the final phase of a solution. After all, a modification of the Cage method has been used even at solving of the 128x128x128 Rubik's cube (scroll down to the end of the linked page).