In the Roman calendar before Caesar, a year consisted of 12 months, for a total of 355 days plus an intercalary month between February and March.
For the Romans, the ideal intercalary cycle consisted of ordinary years of 355 days alternating with intercalary years (377 and 378 days long). On this system, the average Roman year would have had 366¼ days over four years, giving it an average drift of one day per year. Later, it was refined so that for 8 years out of 24, there were only three intercalary years, each of 377 days. This refinement averages the length of the year to 365¼ days over 24 years. In practice, intercalations did not occur as they should, according to the whims of the priest in power at the time.
According to Wikipedia: If managed correctly this system allowed the Roman year, on average, to stay roughly aligned to a tropical year. However, since the Pontifices were often politicians, and because a Roman magistrate’s term corresponded with a calendar year, this power was prone to abuse: a Pontifex could lengthen a year in which he or one of his political allies was in office, or refuse to lengthen one in which his opponents were in power. If too many intercalations were omitted, the calendar would drift rapidly out of alignment with the tropical year. Moreover, because intercalations were often determined quite late, the average Roman citizen often did not know the date, particularly if he were some distance from the city. For these reasons, the last years of the pre-Julian calendar were later known as “years of confusion”. The calendar image is a pre-Julian calendar that was part of a fresco.
The problems became particularly acute during the years of Julius Caesar’s pontificate before the reform, 63–46 BC, when there were only five intercalary months, whereas there should have been eight, and none at all during the five Roman years before 46 BC. Thus, Caesar crossed the Rubicon on January 10, in 49 BC of the official calendar, but the official calendar had drifted so far away from the seasons had that it was actually mid-autumn.
Julius Caesar reformed the calendar, and then it was reformed several more times before 1582 (as with Christmas) when the Pope adjusted the solstice to December 21.
Many other cultures celebrate the New Year in a month other than January (which, in truth, is a completely arbitrary date). These include the Mayans, Babylonians, Persians, and Balinese (March); the Assyrians, Nepalese, and much of Southeast Asia (April); the Coptics (September); Australian Aboriginal (October); neo-pagans (November); and in the Muslim world.