LPELL

Last pair and Edges of the Last Layer (LPELL) is a method that solves the last F2L pair and all edges of the last layer simultaneously. There are 48 cases (+ 48 mirrors).

Currently, this is the only full list of LPELL algorithms on the Internet. All of the algorithms are move optimal and the average move count (excluding leading or ending AUFs) is equal to 7.375 HTM.

The algorithms are divided into 8 categories, each covering one of the 8 possible orientation cases. Each category contains 6 possible permutation cases. I have denoted the UF, FR, UB and UL edges with the letters F, R, B and L respectively. These letters correspond to the different faces where each edge is located. The letters below each picture indicate which edges need to be swapped in that particular case, while O means that all edges are permuted correctly.

Some of the algorithms include a U-turn at the beginning which indicates what position the pair needs to be in before the algorithm can be performed. The algorithms do not include any ending U-turns since each case can end in any possible misalignment of the top layer after the algorithm is performed.

Each case is shown with all of the correctly permuted edges correctly aligned with the rest of the F2L. However, around 3/4 of the time this will not be the case since each permutation case can be shifted in 4 different ways in relation to the last pair and the first two layers (the pair needs to be in a certain position, so AUF-ing to align the pieces is not allowed). I advise you to recognize the cases by looking at which pieces aren't placed correctly in relation to each other instead of looking at matching/adjacent/opposite colors. 

All Oriented

                                                                          O - U (R U') R2 U2 (R U R') (U R)        FB - U2 (R U2 R')             FL - U2 R2 D L' B2 L D' R2    FR - U (L' U2) (R U R') (U2 L)     BL - U (R U R' U') (R U' R')            BR - U (R U' R')

Vertical Line

  O - U (R U' R') f (R U R' U') f'           FB - U' F' U' F' L F L' F               FL - U F' (L' U' L) F                FR - R' L F2 R F' U F' U' L'       BL - U2 (F' U F) U2 (R U' R')     BR - U2 (F' U F) U' (R U2 R')               

L-Shape

O - U2 (F' U F) U (R' F R F')     FB - U (F' U' F) U (R U2 R')   FL - U' (F' U' F) U2 (R' F R F')         FR - (R' F R F')                  BL - L F' R' L F' R F L2         BR - (R' F R) U2 y L' U' (L U' L')

Downwards L-Shape

                                                                              O - U2 (R U' R') F' (L' U' L) F          FB - U F' L' U2 L U F            FL - R B' U2 L' B' L U2 B R'       FR - U (R U' R') f' (L' U' L U) f         BL - U R (B U' B') R'                   BR - U2 F2 L F L' F

F-Line

  O - U R B U2 B' U R' (F' U F)        FB - U R U' (B U B') U' R'      FL - U' F2 L2 F U' L' U F' L2 F2    FR - U F B' D R' D' F' B           BL - U2 (F R' F') R2 U2 R'           BR - U2 R U' (B U' B') R' 

L-Line

 O - U R U' R2 U' (F' U F) R      FB - U2 F2 L F L2 U' L U F   FL - U2 (R U' R') U' (R' F R F')   FR - U R B U2 B' R' (F' U F)   BL - (R U' R') (F' U' F) (R U R')    BR - U R' F' U' F U R2 U2 R'

B-Line

  O - U (R U R') U2 (R' F R F')    FB - U F2 U' L' U L2 F L' U' F    FL - R' U' F U R2 U' R' F'       FR - U2 R U2 R2 U' F' U F R       BL - U F2 D' B L' B' D F2        BR - U' F D B' R B D' F'

Dot

 O - U2 F' L' U2 L F (R U2 R')    FB - L F' U2 R' L F R F L2   FL - U' F2 L2 F U' L' U F' L2 F2    FR - U F B' D R' D' F' B       BL - U2 (F R' F') R2 U2 R'        BR - U2 R U' (B U' B') R'