Check the firmware version in the consoles settings page. The process works on any Vita (at the time of writing), so turn off auto updates or use airplane mode to stop your Vita updating. No faffing with the Sony Content Manager. Transfer files to and from your console wirelessly with FTP or wired via USB. Use a £2 adaptor to use a micro SD card on the Vita instead of the expensive, hard to find and unreliable Sony cards. Use USB storage on the PSTV, no need to buy expensive proprietary memory cards. Whitelist modification for PSTV - play every Vita game, not just the ones that Sony allow. Incredible PS1 emulation via Adrenaline, with PSX2PSP versions of games being perfect - a PSone in your pocket! All PSP games can be played, not just the ones Sony whitelisted on the PS Store. Overclocking, graphical filters, and PSP games look incredible on that screen. Incredible PSP emulation with an emulator called Adrenaline. Retroarch, with emulation of all the staples - Snes, Megadrive, Gameboy Advance, Spectrum (who needs that Vega+ anyway?) and so on.
The process for the mod varies depending on the firmware that's currently on your Vita, but it's simple enough, and the console becomes an indispensable travel companion once you've modded. The most recent mod methods allow any firmware to be modded, and you can still go on PSN to download your purchases. The PS Vita has been hacked wide open, and some of the stuff you can do with it will amaze you. This original post has been updated to be current. Please note that many of the issues discussed in the replies to this thread are out of date now.