Solving 4x4x4 and 5x5x5 Supercubes

Although the 4x4 and 5x5 supercubes look much more difficult and impressive than the standard ones (even experienced bigcubers will wonder how you can find pieces so quickly!), they aren't much harder than the normal cubes: all they take is a slightly different method. The following tips and tricks were all gotten through experience and practice. Note that I'm going to describe the reduction method below, for a couple reasons: I think it's the fastest known way to do supercubes (and normal cubes for that matter), I'm most familiar with it, and you're probably most familiar with it. So if you're not a reduction user, then sorry, the only advice I can give you is to go practice :)

Centers

Centers are the only part that's really different about supercubes, although they are not the only part that you have to solve differently. In a supercube, every center has to be solved into the correct spot - they're not interchangeable anymore. Thus, instead of just solving blocks of centers, you have to actually pay attention to what goes where.

In a 4x4, you only have x-centers, which have three colors in the standard Pochmann sticker design (which I really recommend, because just drawing arrows or cutting corners off the stickers makes it very hard to see what goes where). I always put 4x4 centers together by making two 1x2 blocks. This probably doesn't require any explanation at all, although you might find it helpful to know that if you pair up the two yellow centers across a red-yellow 'edge', the other stickers will be paired across an orange-yellow 'edge' (since orange is the opposite of yellow). This can make recognition and setup easier.

If you go to solve centers, you might notice that the last center doesn't automatically solve itself anymore. That should happen 5/6 of the time, but fortunately there are algorithms to fix this quickly. If you want to swap Ufr and Ufl, you can use this A-perm-like algorithm:
x r' U r' d2 r U' r' d2 r2
If you want to do a diagonal swap, on the other hand, the algorithm I currently use comes from the Square-1 and is as follows:
r2 U D r2 U r2 U D r2 U r2 U D r2.
So your centers should be solved and you can now start on the edges step.

In a 5x5, on the other hand, you have many different types of centers, and now the fixed center, although it cannot move, must be oriented along with the rest of them. Again I pretty much solve the centers with my normal method (1x3 blocks, with the first block containing the fixed center) although it takes a bit longer to find the exact pieces I am looking for. If you see any blocks already formed for the first center or two, that will be very helpful, so try to preserve them.

On 5x5 you still have last center problems, but since PLLs are hard to recognize and turning a fixed center (only) by 180 degrees is slow, I prefer the following method. First I solve a 2x3 block on the second to last center, and position it so I can do rUr'-type triggers without messing it up. Then I intuitively solve all of the T-centers on U and F. It's important to solve them relative to the fixed center (not just relative to each other). There is one other case here, that of parity, where you have two edges swapped. The quick solution is just this:
r U2 r U2 r U2 r U2 r.
That algorithm should do an opposite swap of edges on U (a Usl-Usr swap to be exact), so now you can be certain to be able to solve the edges only. Finally, I have six T-centers left; I do an F turn to set them up to a more convenient location, and then finish off the centers with U turns and the Niklas commutator (which has two forms):
r U l' U' r' U l
l' U' r U l U' r'.
Sometimes it is actually more convenient to just solve the F centers with this commutator, and then finish off U with an A perm (which does exactly what you think it does). If you get an E or an H, good luck, you might as well just do two As. The A perms (for 5x5) are:
x r' U r' d2 r U' r' d2 r2
x r2 d2 r U r' d2 r U' r.

Edges

If you got the Eastsheen supercubes, you might notice that your edges and corners have more colors than they need to. The extra colors don't affect anything (although sometimes they can make recognition for PLL a little easier, on 3x3), and after a couple of dozen solves you'll learn to ignore them.

For 4x4, you can do edges exactly as you would on a standard cube. There is no change at all no matter what method you use, so go full speed on this step.

For 5x5, if you use AvG or any other kind of 2-pair edges method, do whatever you normally do. You'll notice that you never get parity - this is because we fixed it during the centers step. Convenient, isn't it? If the T-centers are solved, you can never get parity on the wings. Anyway, if you use the freeslice/bigcubes method, the biggest difference is that during the first 8 edges you can't do half turns of the faces (F2/R2/B2/L2 if you solve in the E slice, for instance) unless the slices are all solved. Just making them parallel isn't enough, because that will actually slightly mess up two centers. You will also notice that, during the last 4 edges, you can't do the m U2 m' U2 algorithm, because that also messes up centers a bit. (If you really want to do it, you should do the inverse algorithm twice.) You should also be careful during the last 2 edges: if you solve this in one step, be very careful about whether your algorithm messes up centers or not, because some will and some won't. If your algorithm does mess up centers, you can always just fix the edges with the r [flip edge] r' algorithm, which is completely center-safe.

3x3x3

The final step on both 4x4 and 5x5 is to solve a supercube 3x3. You don't need to learn any new tricks for this, you just have to be more careful. When you're putting together the cross, make sure that the centers are all oriented the right way; since this takes so much more thought than a normal cross I do not at all expect you to do this in one look. Feel free to do it one or two edges at a time. Now, if the centers on the first two layers are correctly oriented, you are basically home free: almost all F2L algs will keep them solved (the only one I know of that messes them up is the 2-gen edge insert algorithm, but you can always use an alternate algorithm), and most OLL algs will keep them solved as well. Some PLL algs do, and for certain ones (U perm, for instance) it might be helpful to learn alternate algorithms that don't mess up the centers at all, since you have to spend extra time fixing it. Here's one possible alternate U perm:
F2 U (M' U2 M U2 M' U2 M) U F2.
If you do end up with a messed up center, you should twist both it and the U center, because the U center hasn't been fixed to a specific orientation yet anyway. The following (intuitive) algorithm will flip the front center clockwise and the left one counterclockwise (note that here M and E mean every slice together):
M' U M E M' U' M E'.

There are three parity algorithms that you should know. The first two are the standard 4x4 parity algorithms (so of course they only show up on even supercubes). For the orientation parity, you can still use the normal one although that will twist the R or L center by a quarter turn, so I strongly suggest that you use some kind of slice-turn-only parity. My favorite is this one:
2R U2 x 2R U2 2R U2 2R' U2 2L U2 2L' x' U2 2R U2 2R' U2 2R'.
For PLL parity, the normal algorithm will mess up centers pretty badly, so here is one that won't:
x' 2U' R F' U R' F 2U 2D F' R U' F R' 2D' x
Finally, at the very end on both 4x4 and 5x5 supercubes, half of the time the U center will be twisted by 180 degrees. There are two algorithms I use for this, either of which can be faster depending on the cube and your style:
(R L U2 R' L' U)2
(R' U' R U')5.

And with that, you know everything you need to solve the 4x4 and 5x5 supercubes every time. Good luck, and good times!