The Babylonian Calendar
The Babylonian calendar was lunisolar, which means that periodic leap months were required to keep the lunar and solar years in synchronization. The months began at the first visibility of the new crescent at sunset. In later Babylonian times, the new moon was determined not by direct observation but by a complex mathematical rule, which in fact yielded a very close result.
The intercalary month was inserted either after Ululu or Addaru, and it was simply called Second Ululu, or Second Addaru. There is some evidence that by the reign of Nabonassar (747 BCE) Babylonian astronomers had discovered the Metonic 19-year cycle, but until the 4th century BCE, there is no evidence that a 19-year cycle was used to assign fixed intercalary years within the cycle. In its fully developed form, years 3, 6, 8, 11, 14, and 19 had a second Addaru, and year 17 had a second Ululu.
For earlier Babylonian history, years are reckoned by the regnal year of the ruler. After Seleucus I conquered Babylon, scribes began to record dates in the Selucid Era (SE), a continuous count of years that did not stop with the death of Seluceus. Year 1 SE corresponds to 312/11 BCE, a correspondence that can be confirmed by records of astronomical observations dated in this era.
After the Parthians conquered Mesopotamia, the western part of the Selucid empire switched the beginning of its year from spring (Nisanu) to fall (Tashritu), under Greek influence. The Parthians kept Nisanu as the beginning of the year.
The seven-day cycle makes its earliest appearance in Babylonian documents of the 7th century BCE. It is not quite yet the week as we know it, however. In origin, it seems to have been one fourth of the approximate time in a month the moon was visible. In short, it does not include the days around the new moon, and is not therefore a continuous cycle. To picture what this "week" was like, imagine one of our months with four regular weeks, and then a few epagomenal days at the end of the month, which do not belong to any week.