Instead of the decimal system that is familiar to us, the Mesoamerican peoples, notably the Aztecs, used a numbering called vigesimal or of base twenty. The calendar played a fundamental role in Aztec society. The Mexica used a first calendar based on a tropic year of 365 days, called the xiuhpohualli which had 18 months of twenty days, or 360 days, to which were added 5 additional days. Alongside the annual solar calendar, the Aztecs also used a divinatory calendar or tonalpohualli which included 20 periods of thirteen days called “thirteens”, which led to a sacred year of 260 days. The two calendars (tropic and divinatory) took place simultaneously in time. Identical designations for the days in the two calendars occur after a period of 52 solar years or 73 divinatory years. The different hypotheses formulated concerning the origin of the 260-day cycle are discussed in the text. Some have seen this as the number closest (in the vigesimal system) to the length of human gestation. Others, like Sahagún, have claimed that tonalpohualli is not based on any natural or astronomical cycle. The most plausible hypothesis results from the observation of the solar cycle and, in particular, from the observation of the passages of the Sun at the zenith.The Aztecs offered numerous sacrifices, including human ones, to combat dysfunctions in the cosmic order of the world.