Yes and No.

The new PS3 model, 40GB and the upcoming 80GB Core do not have any PS2 compatibility. 20GB and NTSC 60GB PS3 model support almost all of PS2 games. PAL 60GB & old 80GB support ~70%, models are somewhat backwards compatible. They have the PS2’s GPU, but uses software to emulate the PS2 CPU.

All PS3 versions support PS1 games.