Beginning of the 21st Century and the Third Millennium
Years of the Gregorian calendar, which is currently in use, are counted from AD 1. Thus, the 1st century comprised the years AD 1 through AD 100. The second century started with AD 101 and continued through AD 200. By extrapolation, the 20th century comprises the years AD 1901-2000. Therefore, the 21st century will begin with 1 January 2001 and continue through 31 December 2100.
Similarly, the 1st millennium comprised the years AD 1-1000. The 2nd millennium comprises the years AD 1001-2000. The 3rd millennium will begin with AD 2001 and continue through AD 3000.