Meyer Method for 4x4+
The Meyer method was originally developped for solvers using Roux, but since both APB and Roux usually start with a 1x2x3 block, Meyer can be used with APB as well. This tutorial is explained using a 4x4, but Meyer is used on bigger puzzles as well.
This tutorial is written for people who can already solve a 4x4 using other methods.
Tutorial
Solve two opposite centers
As the first step, two opposite centers are solved. One of these centers will be your L-face color for the 1x2x3 block. If you can only solve with a specific block color, make sure to choose the correct centers to solve here.
Create a 1x2x3 block on the bottom-left
After creating the first two centers, you can create a 1x2x3 block on the left side of the cube. To achieve this, you can use block building. If you are struggling to find efficient solutions, try first pairing the DL dedge by bringing both wings to DF and then solving the dedge. Then you can attach two F2L pairs. You can think of the two pairs as regular pairs, just with one more piece.
Solve the remaining centers
Now, you can solve the remaining centers using R-, 2Rw-, 3Rw- and U-moves.
Pair the remaining wings
To pair the remaining wings, you should first do a y' x'-rotation. Then you should first solve any dedge and put it into DF. Now you can pair the other edges using 3-2-3 edge pairing on 4x4 or Freeslice on bigger cubes.
If you insert pieces into back slots during edge pairing, make sure you place an unsolved corner at that slot using D-moves (for example by inserting an edge into BR using D R' U' R D' instead of R' U' R). Otherwise you will destroy your 1x2x3!
Solve like a 3x3
Finally you can undo the y' x'-rotation and solve the cube normally. Your next step is solving the DF and DB edges to complete the 2x2x3.
Dealing with Parity
For cases where you do not recognize that you have parity (on 4x4 and one 6x6 case), you should solve OLL parity before solving edge orientation. In the other cases it should be solved immediately after pairing the wings.
PLL parity should be done during the last layer. If you use OCLL + PLL, you should correct PLL parity before or after PLL, if you use ZBLL, you should correct it before or after ZBLL.
Learning more Optimized Algorithms
Tips on which algorithms to use on big cubes can be found here.