Part 1: In this step you are finally going to solve your cube. What you're going to do is solve your cube as if it were a normal 3x3x3 Rubik's Cube. First of all if you still don't see how your cube is a 3x3x3 then think of it like this
4x4x4 as a 3x3x3 Cube
All you have to do is treat the center groups as one center piece and treat each edge piece group as one edge piece and you will be able to solve your cube the same way as a normal Rubik's Cube. There are two positions that can come up though that you will not be able to solve the same way. There are three things total that can happen in this step. Either your cube will solve exactly like a rubik's cube with no problems, two edges will be switched, or one edge will be flipped and the rest of the cube solved.
Hold the cube as in the front view diagram and do the move r ² U ² r ² U ² u ² r ² u ². This move will solve these two edges in much the same way the move from Step 1 will solve two center rows that are switched.
This is the same position as the one with two edges switched except now the edges are solved and that messes up the corners. Since the position is essentially the same you would fix it in the same way. Hold the cube so the corners that need to be switched are both in the U face and then do the same move as above which is, r ² U ² r ² U ² u ² r ² u ². Now the top face will be semi-scrambled but it is solveable now. Just go back again and solve the cube the same way you would a 3x3x3.
Now hold your cube as in the diagram and do the move r ² B ² U ² l U ² r ' U ² r U ² F ² r F ² l ' B ² r ²
Remember that this move only corrects the orientation of the one incorrect edge group. The last layer of the cube may still be semi-scrambled, and you may still need to finish applying your 3x3x3 solution to solve the cube.
Disclaimer: I often receive e-mails saying that this algorithm is incorrect and that it leaves people's cubes scrambled. I can assure you the move is correctly written. If after applying this algorithm your cube is completely scrambled try to undo what you did. If you are unable to undo what you did, then go back to step 1 and start over. If this move leaves your cube scrambled please visit the notation page and make sure you are reading the notation correctly.
If you're interested in speed solving then you can use the following move which turns more faces but it's much easier to do quickly. This move will mess up both the orientation of some of the corner pieces in the U layer as well as the positions of some of the edge groups. Therefore if you plan on using the speed solve move I strongly recommend you use it at a point in your solution where it doesn't matter that the pieces will be scrambled around a little bit in the U layer. Your edge groups and centers will stay together, but the edge groups in the top layer will move around a little bit. For example I use this move right after I have finished solving the first two layers (I use a layers method as my solution to the 3x3x3) when there is an odd number of edges showing the correct color up. If you use this speed solve move, be aware that it messes up the U layer a little bit.
Ok, so here's the move. Each group of moves in parenthesis can be done at the same time. (R ² r ²) B ² U ² (L l) U ² (R' r ') U ² (R r) U ² F ² (R r) F ² (L ' l ') B ² (R ² r ²).
The first move listed in this section will flip only the two edge pieces in UF and leave the rest of your cube solved. The second one is for if you want to do the move quickly, but beware that it messes up the U layer a little bit.
3. These two extra cases occur at exactly a fifty-fifty chance for each one. Also don't be surprised if your cube has a combination of both of these extra cases, with two edges switched and one of them is flipped and the rest of the cube is normal.
Now your cube should look like this,
Congratulations! You have now solved your Rubik's Revenge! After doing all the moves a few times you'll get to the point where they will all make sense and you will be able to solve the Revenge every time you pick it up. Now that you can solve the Revenge if you get into speed solving then you can submit your times on my Unofficial World Records Page.
If you found any problems with my solution or any confusing explanations then please let me know so I can clear them up. IMPORTANT
If, after going through my solution and doing the moves exactly as I have them and your cube comes to a weird postion like only one edge piece is flipped not a whole edge group or only one corner is flipped clockwise or counter-clockwise then that means your cube needs to be disassembled. For instructions on how to disassemble your cube click here.