You can check if your games are compatibility here on this page. The latest version supports 1697 games, including all time favorites like Shadow of the Colossus, the God of War and Grand Theft Auto series, Kingdom Hearts and of course Final Fantasy and Metal Gear Solid. One of the best Playstation 2 emulators has recently been released as version 1.0. If not, your best solution to play them besides connecting your old PS2 to the television is emulation.
If you still have old Playstation1 or Playstation 2 games you may be lucky enough to play some of them on a PS3 system if you own it. It was the time when Microsoft entered the market with its first Xbox, which I modded into the perfect multimedia device. The Playstation 2 was the dominant system of its time, and even though my beloved Sega Dreamcast was ahead of its time in many regards, it was the Playstation's general appeal and the success of the Playstation 1 that made it the number one system back then.