The AI method - 4x4x4 Tutorial

Introduction

The AI reduction method is basically the reduction of a 4x4 to a 2x2. This way of reducing implies no parity. The basic AI concept and the reason for this lack of parity is explained in this article.

There are actual AI Cubes on the market, but this is not necessary for this kind of reduction. You can use a standard 4x4x4 cube and reduce the first two layers using the layer by layer method or as 4 corner blocks. You can apply Step 1 four times to get the latter done. After this is done (if you don't have an AI Cube) you can start with the tutorial below. 

Furthermore, the only 2 things you need to know before starting this tutorial is (1) how to solve a standard 2x2x2 and (2) these two algorithms: 

Rotating corner block: (R' D' R D) (R' D' R D)
Corner swap: (2R U 2R U' 2R) U' D (2R U' 2R U 2R) D'

These algorithms should be executed by pretending you're dealing with a 2x2x2, thus turning two layers.

The cube used in the pictures is a regular 4x4x4. To clarify: The 4 corners blocks I'll reduce has orange on it (meaning the 2 reduced layers, or the 4 reduced corner blocks, at the start, have red on the bottom). The tutorial is based on the pictures because there're many ways to reduce corner blocks and there are many cases for the last one. I'll explain a general approach for the latter when we get there and there's also a video below in which I show some examples of the reduction of the last corner block.

If there are any obscurities, you can simply comment below. Good luck!

Tutorial

Step 1 - First corner block

As you may expect, the first corner block is the easiest, for there's many workspace. The way the block-building works is not that difficult once you've got the hang of it. It might be a good idea to build the first block a couple of times, because the less you need to think about it during the next steps, the better.
There's another thing I might need to mention before you actually begin: when I talk about the upper layer, I mean the top 4x4x4 layer. The bottom layer is the 4x4x4 layer underneath this upper one. In other words, the upper and bottom layer is together the scrambled top half of the whole puzzle. (I don't consider the 4 blocks which are already built at the start as a layer when I talk about layers.)

For the first corner block, select a random corner piece which will define your first block. This is the orange-green-yellow corner in the pictures.
Next, find an edge piece which will match next to the corner. I've chosen the yellow-orange edge. By moving the upper layer, you can match the corner up to its edge if the latter is in the bottom layer.
However, if the edge is in the upper layer, you can move it to the bottom one by using the first algorithm mentioned above (you may have to use it twice to get it down). This algorithm essentially rotates a whole corner block around counterclockwise
If the edge has the right colors, but doesn't match with the corner, that means you've got hold of the wrong one. There are two yellow-orange edges; so try the other one if the first one doesn't match.
Once you've got this corner-edge pair, move it to the upper layer by performing the first algorithm again (this way it doesn't get broken if you move the layers independently).

The next thing you need is another edge which fits to the corner, I use the green-orange edge in the pictures. Once you've assured yourself that you've found the right edge, match it to a center piece so it can form a 2x2x1 sheet (I'll keep calling this a sheet) with the corner-edge pair you've already made. In my case (in the pictures), this center piece is orange. You can match the edge and the center the same way you've matched the corner to the previous edge.
Place this edge-center pair in the bottom layer with the same algorithm and match it to the previously built corner-edge pair to form the 2x2x1 orange sheet. Place this sheet safely in the upper layer using that same algorithm.

Now we'll make a base which is essentially the remaining edge that fits to the corner (in my case; the green-yellow edge) matched up to two center pieces (green and yellow in the pictures).
To make the base, just use the same algorithm to put center pieces in the bottom layer and rotate the edge.
Now you can easily match the sheet to the base by placing the base in the bottom layer.
Congratulations, you've made your first corner block!

Step 2 - Second and third corner blocks

The second and third corner blocks are made in exactly the same fashion. You can follow the steps from Step 1 to assemble the next two corner blocks. There's one thing, however, to keep in mind. If you turn the upper layer, you should always remember to turn it back because you're breaking up the first block.
For example, say you want to match an edge to a center and the center is already in the bottom layer. You can move the upper layer to match them up, then place them as a pair at the bottom/top layer and move the upper layer back to restore the corner block. This shouldn't cause any problems though, it's just something to keep in mind.

There's a difficulty you can run into when you want to match a sheet to its corresponding base. You cannot simply move the top layer to match them up, because you'll break you're first (and maybe second) block. There is, however, a simple way to solve this problem:

Hold the cube in such way that the base is in front of you in the upper right corner block and the sheet in the upper left corner block. (In the picture: we want to match the orange-green-white sheet to the orange-green base.)
Now turn the upper layer twice so the base is in the back at the right and perform the second algorithm. Now the sheet is matched to its base (with no blocks broken) when you move the upper layer once more counterclockwise.
With this in mind, and the steps from Step 1, you can build the next 2 corner blocks.

Step 3 - Last corner block

The last corner block isn't that easy and requires some insight in the first algorithm. The first picture shows the pieces of my last corner block, the yellow-blue-orange one.
An important rule to keep in mind with this last step is: make as much bars as possible. With bars I mean corner-edge and edge-center pairs.
In my case, I already have two bars, the corner matched to a orange-yellow edge and the blue-yellow edge matched to a blue center piece.
The reason I don't consider this edged matched to the yellow center piece as a bar, is because the edge matched to the blue one belongs to the same sheet as the corner-edge pair, meaning I can put them in the upper layer and conserve them both at the same time (think of our rule!).

So the next logical step is to put both pairs as one sheet on top, as shown in the second picture where the whole corner block is rotated.

This step is hard to explain in text and is very unpredictable for each solve, so I surely recommend watching the video below where I show a couple of last-block-cases. The whole point, is to let every piece be part of a bar. This means that matched up center pieces from other blocks must be re-matched to other edges. (You must break at least one already-made corner block in this step.) Following this logic, it can help to put a corner block with the color(s) you need (in my case: orange) next to your last corner block (I got lucky and didn't need to swap blocks) using the second algorithm. In the picture below, I've lent a yellow-orange edge and a yellow centerpiece from the green-orange-yellow corner block to its right.
This last step takes some fiddling around and is most simply approached intuitively. It's a lot of trial and error and some thinking ahead.

You always have to remember our rule and try to conserve every bar you create, especially bars which include a corner piece!

Once all the pieces are part of a bar, you can start matching them up to form sheets and bases using the same technique as before.
Once all the bars are part of a sheet or a base, you can match those up using the second algorithm which is explained in Step 2.
Once every sheet and base are matched up, you'll end up with a regular 2x2x2! Now you can solve the whole puzzle as you would solve a 2x2 with whatever method takes your fancy.

Conclusion

I find this an interesting way of reducing a 4x4 because there's a lot more to it than the standard way of reducing the puzzle to a 3x3. The first 3 blocks are rather intuitive, while the last one requires some more insight and planning ahead. This might be difficult the first couple of times you approach your 4x4 like this, but the more you do it, the easier it becomes and the more you can plan in your head how the different bars can be formed, placed and matched up.

I highly recommend solving your 4x4 like this because it opens a whole world to a lot of new possibilities to solve your puzzles which need reduction. If you've mastered the 4x4, the 5x5 AI reduction shouldn't give any problems. And with a little bit of thinking, you could solve the 6x6 the exact same way!
Non-cubic puzzles can also be approached using this method! I'll post a tutorial on the Gigaminx AI solve very soon, but I surely recommend trying this for yourself first, because the Gigaminx correlates to a 5x5x5, which is a simple extention from a 4x4 AI solve.
Either way, enjoy solving and don't be afraid of some (new) challenges.