Lunar cycles are varied and extremely complex and yet the moon has more effect on the earth than any other body except the Sun. Not only are ocean tides important in shaping the earth, and are affected more by the moon than the Sun, but tides in the air are important for determining the weather which in turn affects so many other variables from plants and crops, to animals and the economy.
The simplest cycle of the moon is the daily rising and setting which gets later each day by about 49 minutes and this is the major component of tides which occur twice per day. Therefore tides move through each hour of the day twice in every 29.53 day lunar phase cycle.
In modern times many countries keep a solar calendar that arbitrarily fixes the year at 12 months, but in fact there are 12.368 months in a year and so calendars that use lunar months must insert an extra month about every 2.72 years. This 2.72 year cycle brings lunar phases back to the same time of the year, but because 2.72 is not near an integer it only does so approximately and more similar lunar phases at the same time of year occur after 8, 11 or 19 years.
The word "month" may be used to mean many different things, with the calendar month of over 30 days being the only one not related to the moon. The phases repeat in 29.53 days while relative to the stars the moon goes around the earth in 27.32 days.
Things would be simple enough if that were all the variations, but the moon's orbit is tilted 5 degrees to the ecliptic and precesses in a cycle of 18.6 years so that the moon's orbital tilt to the earth's axis varies from 18 to 28 degrees over this period. This variation is important for tidal effects because when the moon is above a higher latitude on earth the once per day tidal component becomes stronger relative to the twice per day component especially at high latitudes.
Because of this wobble of the moon's orbit, there is another month called the nodical month which is 27.21 days, being the time between crossing the earth's equator going in the same direction.
One last major variation is the rotation of the long axis of the ellipse within the wobbling orbital plane. This takes 8.85 years to travel around the earth, or 6 years relative to the wobbling plane.
The returns of the moon to the closest point to the earth is another month, called the anomalistic month which is 27.55 days long.
Because two 8.85 year anomaly cycles makes 17.7 years and this is a bit less than the 18.6 year nodal cycles, there is a gradual drift between these two interacting which only repeats after 180 years or 90 years if the two nodes are regarded as equivalent.
As was mentioned the 18.6 year cycle is important in determining the weather as is half of this, or 9.3 years. These cycles can be found in crop yields and in geological formations. However the moon is gradually receding from the earth which changes all of these periods very slowly. Professor Afanasiev of Moscow University has designed a method that he calls "Nanocycles method" of very accurately dating geological formations by finding the period which is presently 9.3 years and its interaction with the seasons. The 9.3 year cycle comes at the same time of year on average every 31 years because 9.3/.3 = 31. The nearest repeat of the seasons will actually happen after 28 years 2/3 of the time and 37 years 1/3 of the time. However this 31 years cyle of seasonal interaction of the is very sensitive to small changes because when the cycle was 9.2 years the interaction was in 9.2/.2 = 46 years. Professor Afanasiev has used this to accurately date deposits and so determine other geological cycles very accurately.
Eclipses depend on three things coming together at the same longitude, the Sun, the moon and the moon's node (either one). For a lunar eclipse the moon is opposite the Sun while for a solar eclipse it is on the same side. Because the different months have to come together there are certain intervals after which eclipses tend to repeat. This repetition happens after 18.03 years and is called the Saros cycle.