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~~ How to fix the 4x4x4 solution ~~
~~ All the animations start over on the grey cube ~~
~~ Complete move notation on bottom of page ~~
Fix all the corners using any 3x3x3 cube method you might know. You can either fix the left and right centers now or wait till the end.

Fill in all the edges on opposite sides (left
and right) using a move like U r U' or U l' U' keeping one top slot open, this enables you to fill in all opposite side edges except one.
(red is the left face in animation)

~~ Last Outer Edge ~~
To get the last edge in place...
turn the slice/spot where the edge is supposed
to go facing up, and get the cube itself on the down slice with
the same color facing outwards that would be the correct color if it were
on the up slice facing upward
(that's yellow in the animation)
Use the moves ...U D2 r' D2 r U'
This is the same as moving the "space" where the cube has to go directly above where the cube is now, then giving the bottom a 180 degree turn which moves the inner slice one over (out of the way), then moving the "space" down to the bottom, then giving the down slice a 180 that sticks the cube into the "space", then just turn the "space" (that now holds the correct cube) back up to the top and twist the up slice one turn and you have it, if you can remember the way it works you do have to memorize any moves...
you can see/know what you have to do. 
If the cube is in this type position use the move U D2 l D2 l' U'
notice it's the same type move... one uses right inner, one uses left inner 
(this animation is looking at
the cube from the back (blue),
front view didn't show enough)

~~ Last Inner Edges ~~
This is a way to work the inner slices to get inner edges in place, it's a move that will rotate three edges without mixing anything else up, it will take whatever edge cube is in the...
[left upper front] and send it to [right upper back]
[right upper back] to [right down back]
[right down back] to [left upper front]
Keep anything that might be correct already in the right front or left back and get a cube [left upper front] you want to send to the other slice [right upper back] in the left front top space then do the move
l D2 l' U2 l D2 l' U2
There isn't a flip move for the 4x4x4 edges... if a cube looks like it's in the right place but it's twisted (flipped), it actually means it's not on the correct inner slice. So if you have a pair of edges that look like they need to be flipped... just make sure to put one in the [left upper front] and the other in the [right lower back] positions then do the move. There are only eight inner edges and this moves switches three of them around... you should only have to do this move (at the most) 2 or 3 times and have them fixed (when you use it efficiently). Don't forget... you can move anything you want into a different position before you do this move then undo the different position after, for instance to permute three edges on the same slice just give the "back" slice a 180° twist before and after the move. If you have all the cubes on their correct inner slice and give it a little forethought you should only have to do it once to get then in their correct positions also. This move sends the...
[left upper front] to the [left down back]
[left down back] to the [left upper back]
[left upper back] to the [left upper front]
B2 l D2 l' U2 l D2 l' U2 B2
it's the same move as above with the fore and aft 180's included.
If you want to send [right upper front] to [left upper back] do the move...
r' D2 r U2 r' D2 r U2
If you notice on the third move of either routine, the "up" slice is in the same state as it was at start except for one cube...
I call this "isolating the cube". We have dropped/turned the right or left up cube on the inner slice to the down side, then 180'd the cube out, then turned the inner slice back up, now we can turn the up slice any amount we want, undo the routine, then undo the any amount we turned the up, and have a three cube switch.
Twist the cube/space in question...
turn down, slide out, turn up, rotate switch
turn down, slide in, turn up, rotate switch
You can use either of these above moves any time you want, if a three edge rotation doesn't happen to be lined up exactly as you need it, you can move/switch a cube into one of the three spaces, then do the 3 edge permute, then undo the switch move, this enables you to swap three edges anywhere you want without mixing anything else up. Here's one possibility...
twisting the (down) slice before, then undoing it after...
(D) r' D2 r U2 r' D2 r U2 (D')

~~ Centers ~~
This animation runs back and forth
This is a way to fix any centers...
Also using the "isolating the cube" method.
It will swap centers without mixing anything else up. r' d r U' r' d' r U
Drop/turn any cube from top center to the middle layers, then move the other cube that was on the middle layer into that space/position, then bring the slice back up and turn the up slice until the center you want to change goes into the isolated space (the switch), now we can undo the routine, drop cube to middle, turn middle cube/slice back in, turn inner slice back up, then undo the amount you turned the isolated space.
You have to line up the cubes you're going to switch on a mutually adjoining face, and the cube on top you use for the initial line up can't be the one you want change, the one you're going to change is the one you're going to turn into that space after you've done the first half of the isolation routine. If you want to swap four centers (two front, two back) from the front to back face... hold the cube so the centers are on the front and back face and also on the " r " inner slice, you can use... r2 d2 r2 d2 (note: the centers that were on the " r " slice actually end up on the opposite " d " slice, but that's irrelevant) ...too bad there isn't a c3p0 move or we could call those two "Star Wars"
Here's the move used in animation to go from the red and yellow centers swapped to finished...
r' d r U' r' d' r U
Just like above
down, out, up, switch
down, in, up, switch

Here are the some other center line up possibilities,
"x" marks the spot where you figure out the initial lineup

~~ 2x2x2 3x3x3 and 5x5x5 stuff ~~
If you know how to fix a 3x3x3 cube and learn this 4x4x4 method, you'll be able to fix anything including the 5x5x5 and whatever else they can make, All the moves are the same. The picture on the right shows a possible position on the 5x5x5, you can already see how to do it, correct? The best part is you don't need to memorize hundreds of moves, you'll just know how to do it.
All corners are the same.
All outer slice edge moves can be used on any cube.
All inner slice edge moves on the 4x4x4 can be used on the 5x5x5.
All center cube moves the 4x4x4 can be used on the 5x5x5 (except the 5x5x5 dead center, that's the same as the 3x3x3 center).
Any 5x5x5 edges that are also on the center slice are the same as the 3x3x3 (there are 12).
~~ 3 Corner Permute ~~
Here's a three corner permutation that works on any cube and won't mix anything else up (with this and the 2 Corner Flip move below you can fix all the corners on any cube),
R' D R U' R' D' R U looking at the front... it takes the...
[left bottom] and sends it to the [right top],
[right top] to [left top],
[left top] to [left bottom]. It's the same Isolation move.
~~ 2 Corner Flip ~~
here's a two corner flip that works on any cube and won't mix anything else up ... R' D R D' R' D R U' R' D' R D R' D' R U
if you notice it's the same type Isolation move... take a cube down, flip it around then bring it back up (isolation state), make the switch on the "up" slice then undo everything in reverse.
it's easier to see like this...
R' D R D' R' D R U' (isolated)
R' D' R D R' D' R U (done)
just make sure you have the colors to be flipped to the top facing toward you at the moment (in this case "yellow" on "front" face is going to flip to the "up" face). If you can't get two of the same colors facing you... do the move anyway, then you should have a workable position
~~ 3 Edge Permute & 2 Edge Flip ~~
When the cube is in the isolated state you can rotate the "up" slice any amount you want to change either the position or rotation/flip of whatever cube you like, this also works for edge permutation and flipping, like so...
If you want to permute three edges on any center slice you can use the move...
U2 cL U2 cR
...it's a move that will swap three edges without mixing anything else up, it will take whatever edge cube is in the...
[middle upper front] and send it to [middle upper back]
[middle upper back] to [middle down back]
[middle down back] to [middle upper front].
If you wanted to swap three edges that are all on the "up" slice at the moment... just do the switch first by bringing any cube you like to the "down back" spot, for instance, bring the one that is either on the right or left to the bottom back by using the move...
L2 D' then the 3 edge move... U2 cL U2 cR then undo the switch using the reverse
D L2
Here it is all in one piece L2 D' U2 cL U2 cR D L2
The isolated two edge flip move is as follows...
R' mU R2 m2 R' U2 (isolated) R m2 R2 mD R U2
Take a cube to middle slice, flip it around then bring it back up (isolation state), make the switch on the "up" slice then undo everything in reverse.
~~ Other "CornersFirst" Moves (will mix other edges) ~~
Five White "X" on bottom side.
Two blue up, 2 blue away on other.
R U2 R U2 R U R
Five good corners, keep 'um left, Four down one up.
R2 U' R2 U R2 CC* R2 U R2 U' R2
"CC" rotate entire cube clockwise looking down from top.
Rightside becomes front, Backside is now Right.
Two good corners, keep in front.
R2 U R2 UWBTC* R2 U R2'
"UWBTC" rotate top slice clockwise looking down from top.
rotate bottom two slices clockwise looking up from bottom.
Rightside top becomes front, Bottom two left are now front.

~~ Notation ~~
U u D d R r L l F f B b
Up Down Right Left Front Back
Capital letters stand for the outer faces such
as the front face, back face, down face, etc.
Lower case letters stand for the slices just
behind the outer faces, such as the inner front slice, inner back slice,
inner down slice, etc.
~~Any letter by itself means a clockwise move~~
~~Apostrophe' symbol next to the letter denotes
a counterclockwise turn~~
R : turn the right face clockwise.
f : turn the inner front slice
clockwise.
R' : turn the right face counterclockwise.
R2 : turn the right face
twice, either two clockwise turns or two counterclockwise turns, however
you want to look at it. Whenever making a move, do the move as if you were
looking at that face. For example the move...
B' : would be done as if you were looking
at the back face.
d' : would be done as if you were looking
at the down face.
~~ Middle Slice Notation (3x3x3 5x5x5 only) ~~
Any center slice (between "R" and "L") or middle (between "U" and "D") will have a lower case "c" or "m" then face letter or a "2", meaning turn the middle or center the same clockwise direction you would have turned the adjoining face that was named...
cR : turn the vertical center slice (between "R" and "L") in same clockwise direction you would turn the right face .
m2 : turn the horizontal middle slice (between "U" and "D") twice
mU : turn the horizontal middle slice in same clockwise direction you would turn the "up" face .
mD : turn the horizontal middle slice in same clockwise direction you would turn the "down" face .

