Years of the Gregorian calendar, which is currently in use today, are counted from AD 1. Thus, the 1st century comprised the years AD 1 through AD 100. The second century began with AD 101 and continued through AD 200. By extrapolation we find that the 20th century comprises the years AD 1901-2000. Therefore, the 21st century began with 1 January 2001 and will 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 began with AD 2001 and will continue through AD 3000.
Many initial epochs have been used for calendrical reckoning. Frequently, years were counted from the ascension of a ruler. For a calendrical epoch to be useful, however, it must be tied to a sequence of recorded historical events. This is illustrated by the adoption of the birth of Christ as the initial epoch of the Julian and Gregorian calendars. This epoch was established by the 6th century scholar Dionysius Exiguus who was compiling a table of dates of Easter. Dionysius followed previous precedent by extending an existing table (by Cyrillus) covering the period 228-247, reckoned from the beginning of the reign of Emperor Diocletian. However, he did not want his Easter table "to perpetuate the memory of an impious persecutor of the Church, but preferred to count and denote the years from the Incarnation of our Lord Jesus Christ." To accomplish this he designated the years of his table Anni Domini Nostri Jesu Christi 532-550. Thus, Dionysius' Anno Domini 532 is equivalent to Anno Diocletiani 248, so that a correspondence was established between the new Christian Era and an existing system associated with historical records. What Dionysius did not do is establish an accurate date for the birth of Christ. While scholars generally believe that Christ was born a few years before AD 1, the records are too sketchy to allow a definitive dating.
Given an initial epoch, one must consider how to record preceding dates. Today it is obvious that a year designated 1 would be preceded by year 0, which would be preceded by year -1, etc. But since the concept of negative numbers did not come into use in Europe until the 16th century, and was initially only of interest to mathematicians, its application to chronological problems was delayed for two more centuries. Instead, years were counted from a succession of initial epochs. Even as Dionysius' practice of dating from the Incarnation became common in ecclesiastical writings of the middle ages, traditional dating practices continued for civil purposes.
In the 16th century Joseph Justus Scaliger tried to resolve the patchwork of historical eras by placing everything on a single system. Not being ready to deal with negative year counts, he sought an initial epoch in advance of any historical record. His approach was numerological and utilized three calendrical cycles: the 28-year solar cycle, the 19-year cycle of Golden Numbers, and the 15-year indiction cycle. The solar cycle is the period after which week days and calendar dates repeat in the Julian calendar. The cycle of Golden Numbers is the period after which moon phases repeat (approximately) on the same calendar dates. The indiction cycle was a Roman tax cycle of unknown origin. Therefore, Scaliger could characterize a year by the combination of numbers (S,G,I), where S runs from 1 through 28, G from 1 through 19, and I from 1 through 15. Scaliger first stated that a given combination would recur after 7980 (= 28 x 19 x 15) years. He called this a Julian cycle because it was based on the Julian calendar. Scaliger knew that the year of Christ's birth (as determined by Dionysius Exiguus) was characterized by the number 9 of the solar cycle, by Golden Number 1, and by number 3 of the indiction cycle, or (9,1,3). Then Scaliger chose as this initial epoch the year characterized by (1,1,1) and determined that (9,1,3) was year 4713 of his chronological era. Scaliger's initial epoch was later to be adopted as the initial epoch for the Julian Day numbers.
We would say that Scaliger's initial epoch was 4713 BC or -4712. In the historical system of dating, AD 1 is preceded by 1 BC; there is no year 0. In the astronomical system, AD 1 is designated +1; this is preceded by year 0, which is preceded by year -1. The historical system was introduced in the 16th century. However, the astronomical system was not introduced until the 18th century.