There's also a 5 dimensional one and links to all kinds of crazy.

Interestingly, solving higher dimensional cubes isn't more difficult from an algorithm view. Understanding what you're looking at might be a different story!

There used to be a solution online called the "Ultimate solution", the web page I got it from is now sadly defunct. It showed you how to solve the cube with only two moves. The first was a simple commutator, of form FRF'R' and showed how you can use just this one move to get all the edges in place.

If you then place two of these moves together with a similar structure, you get a move that cycles corners. This is of the form (X)U(X')U', where X is an edge commutator like FRF' (example (FRF')D(F'R'F)D'). This one move properly applied will move 3 corners without disrupting edges.

So, the trick to the fourth dimension is to learn a move on the order (C)W(C')W', where C is the inner part (no final move) of the corner commuter above. And so it goes, each dimension only requiring one more move in the same fashion.

Of course, figuring out exactly how to make that move with the interface is the hard part. And the fun.