PSP

If you have a newer model PSP (slimmer), then you're in luck. Just record the output like you would for a console.

If you have an older model PSP, then you're going to need some extra equipment. There are two options: direct and indirect recording. The direct feed method is similar to how you would record on the newer PSP or other consoles and is preferred. The downside is that it costs more and requires taking your PSP apart and thus voiding the warranty. Indirect recording is essentially a hack: a fixed, mounted camera that attaches to your PSP and records what it sees. This is admittedly much better than the typical DS solution, but it can't compete with direct recording.

For direct recording, the de facto device is the PSP2TV from Team XTender. Take a look at this review and guide to setting it up and some more information.

Some devices for indirect recording are the Innovation PSP TV adaptor, Nyko Play On TV adaptor, PSPonTV, and so on. Chris 'Satoryu' Kirk used the Innovation adaptor for his Mega Man Powered Up runs, if you want to check out results.