[QUOTE=Bosda Di’Chi of Tricor]
The Roman gods were worshiped until Constantine, & then things began to fade for them.
Eventually, it became an illegal faith.
However, small cults, especially those devoted to Artemis, continued into the 18th Century CE, in Italy.
[/QUOTE]
I would quarrel with this last statement, but from a more neutral perspective let’s just say that authorities differ on (1) how authenitic the original data so alleging was and (2) the interpretation of that data.
In more general terms, both the Greek and Roman gods are, for the most part, continuities of deities that can be reconstructed for the common Indo-European religious system (some parts of which survive, albeit in a dramatically different evolved form, as Hinduism). So they go back, in one form or another, to about 4000 BCE if not earlier.
These were polytheistic systems without a single central religious authority (for Greece and for early Rome, anyway), so there was undoubtedly a variety of practices. I can recommend books on the subject though I’m afraid I don’t know any good websites. Try Ken Dowden’s Religion and the Romans and Walter Burkert’s Greek Religion.
As far as survivals, it is generally agreed that many pagan practices survived into Christianity. Whether they were understood as pre- or non-Christian by practitioners is one really sticky point; whether we should now analytically understand them as non-Christian is another sticking point. The institution of Pope has its origins in the pre-Christian Roman pontifex maximus, the top flamen of Rome; does that mean the Pope is Pagan?
The old gods continued to be important to Christian writers and artists right up until modern times, usually as allegorical figures, but there is evidence that at least some of the pagan philosophers also saw them this way. It certainly didn’t survive as an independent religious system in Greece or Rome much beyond the fourth or fifth century, though there may have been pockets in rural areas as late as the sixth. Related Indo-European pagan systems lasted much longer in northern and eastern Europe.
There is a contemporary religion, Neo-Paganism (to use the broadest term), for whom it is very important to believe that they are engaged in a revival of ancient practices which did survive, albeit tenuously, as folklore and secret practices in the keeping of European women. The claim is unlikely but unprovable, and personally I’d like it to be true. But this means that the OP’s question is just as fraught in some quarters as the evolution / creation question is in heartland America these days, and you should be prepared to “scrutinize with an intense scrute” the claims of all sides. My own bias is as a mythologist who studies the traditions from an academic perspective.