backward compatibility for the PS3 with PS2 games is as follows on NTSC systems:

20GB (4 USB slots, no longer in production) – yes
60GB (4 USB slots, no longer in production) – yes
80GB with 4 USB slots (no longer in production, bundled with MGS4 or Motor Storm) – yes
40GB (2 USB slots, no longer in production) – NO
80GB with 2 USB slots (no longer in production) – NO
160GB (2 USB slots, no longer in production) – NO
120GB/slim (2 USB slots) – NO
250GB/slim (2 USB slots) – NO
any other up coming PS3′s – NO

PAL systems only have one backwards compatible system, and that’s the 60GB.

in short: if it has 4 USB slots, yes. if it has 2 USB slots, no.

all PS3s play PS1 games.

http://en.wikipedia.org/wiki/PlayStation_3#Model_comparison

the download that others are referring to is for the PS3s that already are backwards compatible. downloading it to a non-backwards compatible PS3 will do NOTHING but take up space on your hard drive! it will NOT make your PS3 play PS2 games!!!

"If a PLAYSTATION 3 is not backward compatible, you will not be able to add this function through any hardware (i.e., changing the hard disk drive) or software (i.e., updates, homebrew) upgrades or modifications. "
that came DIRECTLY from Sony’s own website: http://playstation.custhelp.com/cgi-bin/playstation.cfg/php/enduser/std_adp.php?p_faqid=232&p_sid=1P9bCDwj

and no, not all PS2 games or PS1 games will work. the 20GB and 60GB come the closest to being fully backwards compatible with PS2 games, but they still do not play 100% of those games or PS1 games. if you buy a backwards compatible PS3 and want to know which of your PS2 games work, check this link: http://www.us.playstation.com/Support/CompatibleStatus