Comment by Oreb

That's not the only way to solve big cubes, but it's indeed the most common way (known as "reduction"), and what most people naturally come up with if they try to solve 4x4x4 or bigger on their own. In addition to what you said, there is also the issue of parity (basically, when you reduce a 4x4x4 to a 3x3x3 by solving centers and edges first, you will often end up with a 3x3x3 cube in an unsolvable state, and you need to figure out some tricks to convert it to a solvable state), but if you know how to solve parity problems on a 4x4x4, you can do it for a cube of any size.