Lunar cycles were the most common basis for early calendars. These calendars were often not based on any mathematical determination of the synodic month at all, but relied, particularly in their earliest formulations, upon direct observation to determine the new month. So when do you define the month as beginning? The two obvious starting points are the new or the full moon. Since the astronomical new moon occurs appears so close to the sun as to be invisible, this gives us three observational possibilities: the full moon, the first crescent of the ascending moon after the new moon, visible at sunset, or the last crescent of the descending moon, visible just before sunrise. There is a correlation between when the next day is considered to begin and which observational choice is made. For a society that begins the new day at dawn, the first day the waning moon is invisible just before sunrise is a good time to start the new month. If the next day begins at sunset, e.g., the Jewish Sabbath, the first observation of the new moon makes more sense. A full moon observation would seem to imply a midnight start. Such observations can be used in either a lunar or a lunisolar calendar. It will result in months that generally alternate between 29 and 30 days, but because the synodic month is not exactly 29.5 days, the alternation will not be completely regular.
Direct observation does have its drawbacks. Clouds might obscure observation, and in a large civilization, there is a problem with ensuring each location stays in sync with the rest. The historical development of lunar and lunisolar calendars is largely a matter of replacing direct observation by mathematical formulas and tables that allow the prediction of the month's start without the need for observation.
Lunisolar calendars have the additional problem of keeping the lunar months roughly aligned with the solar year. They generally do this by the insertion of a 13th, intercalary month at periodic intervals. Once again, in a calendar's earliest stages, this intercalation was generally accomplished through decree from a central source. In later times, regular rules were developed.