Whenever this question is asked, there are always a few incorrect people who pretend that it is 31.12.1999, but bobbobley is correct - it is 31.12.2000. The reason is because there was no year zero, so the counting started with 1 and went up to 100 before starting again at 101.
Although oddly enough, the phrase "turn of the century" originally meant the point at which a century is half-over, i.e. 1 January XX51. But that sense appeared to suddenly die out early in the 20th century to be replaced by the meaning it has today.