Tripod

Tripod is an extremely block-building method, somewhat along the lines of Heise but less complicated, easier to learn, and arguably faster (though not as fast as Petrus). It's pretty fun to use, and is very good practice.

There's already a site about this method (by Jack Eisenmann) out there, but this approach is a bit more advanced, and more a guide to the details of the method (and how it would be performed) than a tutorial.

This is the same as the first step of the Petrus method, and Petrus's website has a better explanation than I do. But basically, you will be solving one corner and the joining three edges. If you are color neutral (and you should be for this method), there are 8 possible blocks to build, so you have plenty of options.

This step is also the same as a step from the Petrus method, specifically the second step. You will be adding on a 2x2x1 block (composed of one corner and two more edges) to the existing one. There are 3 possible ways to do this.

Build another 2x2x1 block, which will restrict your movement even more. If you need help, there are some tips in step 4a of the Petrus method, although keep in mind that your edges won't be oriented here. There are four possible blocks here.

Build yet another 2x2x1 block. You have very little freedom now - if you position the last full layer on top, and the open slot in the front-right corner, the basic moves are R U^x R', R' F R F' / F R' F' R, and F' U^x F. There are some more tricks to learn, though, so see what you can come up with. For the pure Tripod approach there is only one way to make this block, but if you're willing to experiment you can build the block anywhere on the top and leave it misaligned until the end.

The goal here is to pair up a corner and edge, and then solve them so you have only 5 pieces remaining. This is not easy, and there are a few approaches you can take. One is the "frifri" approach, where you only use short sequences which don't disturb the solved pieces, such as R' F R F' . There are six possible four-move sequences to use, and you can always finish this step with some combination of them. Another possibility is to simply memorize algs for each position - there are 71 ways to put in a pair, but since all of those sequences are relatively short, it's not too bad.

Keep in mind that there are three possible pairs here, so you can always benefit from choosing the easiest one.

Finally, the last five pieces. This can be done in one step (thanks to Mirek Goljan!), although there are 107 separate cases. If you want to learn this method for speed, that is definitely the best approach.

My personal approach is simpler, but takes longer. There is one very useful algorithm, which I call the "supertwistflip": U' R U R2' F R F2' U F U' and its mirror U F' U' F2 R' F' R2 U' R' U. These two algorithms flip all three corners and both edges. You can use at most one application of the supertwistflip to get to one of three cases: (1) 2 or 3 corners are flipped; (2) a 3-corner cycle; or (3) two corners and two edges are unsolved. Case 1 and 2 I can solve in one step (note that these are useful algorithms to know anyway, so they're worth learning), and if I get case 3 I will do setup moves to get to one of the PLL cases (always J, R, V, or Y).