Rubik's Revenge is a 4x4x4 version of a Rubik's cube. There are strong similarities between the 4x4x4 version and the traditional 3x3x3 cube, but there are also several important differences. Perhaps the most important difference is that on the 3x3x3 cube the centers always had a fixed relationship with respect to one another. In the 4x4x4 version, there is no such relationship - the middle layer has been replaced by two inner layers. The cube has the following pieces:
Treating 4x4x4 as a 3x3x3
If you already know how to solve the 3x3x3 cube, then it is tempting to use the same techniques on a 4x4x4 cube. The first step is to get the centers correct. This is actually pretty easy since you can mess up the corners and edges as much as you want. Be careful to put the colors in a legal arrangement. This is a problem unique to the 4x4x4 cube since it doesn't have the fixed centers of any of the odd sized cubes. It is very helpful to always keep in mind the colors that are supposed to be on opposite sides. It can also be helpful to place a corner or two as you are working on the centers just to make sure that the colors line up properly.
The second step is to get the edges paired up. I like the following operator since it rotates three edge pieces without disturbing anything else:
edge-3-cycle: r (U' R U) r' (U' R' U)
This will cycle (DFr UFr FRu). If you put one half of an edge-pair in UFl and the other half in DFr, then after the edge-3-cycle, the UF edge pair will be matched. If you are also fortunate enough to be able to move the piece that matches UFr to FRd, then you will get a second matched edge pair at FR. Generally I don't worry too much about matching two pairs at once, I just keep placing matched edges at UFr and FRd and any other unmatched edge at FR. Once you get down to 3 unmatched edges you need to carefully line them up so that the edge-3-cycle will solve all three edges at the same time. If you accidentally go too far an have only 2 unmatched edges, then you'll need to ruin one of the matched ones to create 3 unmatched pairs.
Now the cube can almost be solved like a 3x3x3. As long as you only use face moves, the centers will remain intact and the edges will remain paired. Pick your favorite 3x3x3 cube algorithm and get started. 25% of the time everything will be fine. The other 75% of the time you will run into one or both of the following parity problems:
1) A single edge-pair is "flipped". This is something that never happens on a 3x3x3 - edge flips must always occur in pairs. But in the 4x4x4 cube it is possible to swap two edge pieces, which is the equivalent of flipping the edge-pair. The following operator flips the UF edge pair: (R2 r2 B2) (U2 l U2 r' U2 r U2 F2 r F2 l' ) (B2 r2 R2). The real work is done by the middle sequence (U2 ... l'), which swaps UFl and UBl but has a side effect of exchanging some centers as well (Ufl and Ubl are exchanged with Dfr and Dbr). The prefix and suffix sequences move UFr to UBl and also make the center swaps harmless.
2) Two edge-pairs have been swapped. This is also impossible on a 3x3x3 cube, but happens on the 4x4x4. A nice edge pair swap operator is (U2 u2 R2 r2 U2) r2 (U2 r2 R2 u2 U2) which swaps the UF and UB edge pairs. This operator isn't as complicated as it looks since U2u2 and R2r2 can be done at the same time by turning the upper or right half of the cube. The actual edge exchange happens in the middle r2 move, which swaps to pairs of edges and 4 pairs of centers, but the prefix and suffix insure that the center swaps are harmless.
Depending on your 3x3x3 algorithm (specifically, if you solve corners after edges), you may find that the entire cube is solved except for a single pair of corners that needs to be swapped. This is actually equivalent to case #2 above. Give the face with the two corners a single turn, then you should be able to use standard 3x3x3 moves to cycle the corners as necessary. You will also be able to fix two of the edges, but two edge pairs will still need to be swapped.
The parity fix for #1 is a bit unintuitive, and I prefer algorithms that I understand deeply enough to be able to reconstruct them without memorization. An alternative to fixing the parity problems is to give a single inner layer one turn. This will change the edge swap parity back to even, allowing you to fix all of the edges with the edge-3-cycle. The only problem is that 8 center pieces are now wrong. Fortunately it is pretty easy to move center pieces around without disturbing edges and corners. If you want to swap Ubr and Ubl with Fur and Fdr, you can use:
A Different Solution
I found the parity problems to be somewhat tedious to fix. There's another approach to solving the cube that avoids the parity problems by leaving some inner slice operations until near the end. The process I use is as follows:
The only thing left to do is fix the centers. I usually use variations of