PlayStation Vita is backwards compatible with downloadable PSP games. So, you can play Akiba’s Trip and Monster Hunter Freedom Unite on it. Final Fantasy VII from the PsOne Classics section… eh maybe not.
Sony Computer Entertainment Japan posted a PlayStation Vita FAQ and there is a big asterisk next to the part where it says the handheld is compatible with PSP software downloaded from the PlayStation Store.
It says support for Game Archives (aka PsOne Classics and TurboGrafx-16 games in Japan) are the exception.
In other words, PlayStation Vita won’t be able to run Metal Gear Solid or Chrono Cross. Sony says they will release more information about this soon.
The US PlayStation Vita FAQ says PlayStation Vita will run PsOne Classics. (Thanks to everyone for pointing this out!) The Japanese FAQ states PlayStation Vita will run downloads from the PlayStation Store, however Game Archives, which consists of PsOne Classics and TurboGrafx-16 games, are not included in that statement. That’s… kind of strange since the US FAQ says PsOne Classics will run on Vita. Perhaps, the issue is with TurboGrafx-16 emulation? Sony released a few of these in North America and the PlayStation Vita FAQ in North America doesn’t mention these at all. Siliconera apologizes for the error and confusion.