Is the PS-Vita backwards compatible with PSP games?
Some games I would like to play include Final Fantasy 7: Crisis core, Jak and Daxter: The Lost Frontier, Coded Arms, Tekken: Dark Resurrection, and Secret Agent Clank.
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The Vita can only play certain supported digital versions of PSP games. If your games are on a PSP UMD only then you are out of luck. (Here is a fan made comprehensive list of playable games)
As for the games you listed specifically, only these are available currently on the Vita as a download through the PSN store:
I feel that is worth to point out a homebrew application called Adrenaline. It allows you to "unlock" the PSP Emulator in the PS Vita and PS TV so you can run the full PSP Firmware, meaning that you can run any game that the PSP can (with some limitations).
As an example, here is my PS Vita that has Adrenaline installed with Hatsune Miku Project Diva (JP DL), Lego Harry Potter Years 1-4 (US UMD), GTA Chinatown Wars (US DL):