Yes – like the PS3, the PS Vita supports trophies with compatible games. Some games even include the coveted Platinum trophy. Trophies can be synched to the PSN, and you can view or compare PS Vita trophies on a PS3.
Information about trophies you have earned is saved on the PS Vita system. You can only earn trophies on the first PS Vita system that you start a game on. You cannot earn trophies for that game on other PS Vita systems. If you sign up for PlayStation Network, you can save information about the trophies you have earned on PlayStation Network servers. When you play the game on a different PS Vita system, if you sign up for the same PlayStation Network account, you can earn additional trophies on that system.
You can save trophies for up to 100 games on your PS Vita system, but if you sync your trophies with the PlayStation Network, you can save a lot more trophies.