I'm no expert on Flash games - couldn't make one to save my life, but i've definitely played ones with savegames so it must be possible. I think online flash games use cookies as savegames (often your progress will only be restored if you play the game from the same website); if you're making a downloadable game then you can probably get it to do something more intelligent save-wise.
Edit - From some brief research I think savegames go into .sol files which are stored in a folder that Flash gives every user on the computer
eg: C:\documents & settings\you\macromedia shared files\flash\game47\savegame1.sol
i think.
Edit - From some brief research I think savegames go into .sol files which are stored in a folder that Flash gives every user on the computer
eg: C:\documents & settings\you\macromedia shared files\flash\game47\savegame1.sol
i think.