For ancient Greece, see this older answer. There were as many different ways to count the year as there were Greek states, mostly based on who held certain magistracies or priesthoods at the time. For their convenience, Greek authors of later centuries tended to refer to the cycle of Olympiads; since Olympic Games were held once every four years and were numbered, it was easy to place an event on a chronological scale ("the third year of the 20th Olympiad", etc.). However, there is no evidence that such an overarching system was ever used by either states or ordinary people.