When was the Roman conquest of Greece completed?

That is, by what time were there no more independent Greek states?

It depends what you mean by “independent” and what you mean by “Greek states”. If you’re talking about Greece proper, Rome conquered the Achean League and took pretty much all of Greece itself under its control after the Battle of Corinth in 146 BC. Crete wasn’t conquered until 69 BC. If you’re including Greek and Greco-Persian states in Anatolia, Pontus was defeated in 64 BC, and half of it annexed, but part of it, along with Colchis and Cilicia, stayed nominally independent as a client kingdom of Rome until 62 AD. The Bosporan Kingdom was annexed by Rome in 62 AD, but got independence back in 68 and lasted until 341.

The only thing I would add to Captian Amazing’s post is the fall of Egypt–the last real Greek/Macedonian kingdom–in 30 BCE. This, I believe, is considered the traditional end of ancient Greek culture–though the fall of mainland Greece at the battle of Corinth (146 BCE) is just as good a date.

Thanks, folks! Much appreciated.