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A calendar is a system for naming periods of time, typically days. These names are known as calendar dates. Cycles in a calendar are often synchronised with the perceived motion of astronomical objects. A calendar is also a physical device (often paper) that illustrates the system (for example, a desktop calendar) — this is the most common usage of the word.

As a subset, 'calendar' is also used to denote a list of particular set of planned events (for example, court calendar).

Calendar systems

Calendars can be classified by what their cycles are synchronised with:

A lunar calendar is synchronized to the motion of the Moon (lunar phases); an example is the Islamic calendar.
A solar calendar is based on perceived seasonal changes synchronized to the apparent motion of the Sun; an example is the Persian calendar.
There are some calendars that appear to be synchronized to the motion of Venus, such as some of the ancient Egyptian calendars; synchronization to Venus appears to occur primarily in civilizations near the Equator.
An arbitrary calendar is not synchronized to any external phenomenon; for example the week cycle.
A calendar can also be acyclic. For example, the Julian day used by astronomers is a simple linear count of days.
Very commonly a calendar includes more than one type of cycle, or has both cyclic and acyclic elements. A lunisolar calendar is synchronized both to the motion of the Moon and to the apparent motion of the Sun; an example is the Jewish calendar.

Many calendars incorporate simpler calendars as elements. For example, the rules of the Jewish calendar depend on the seven-day week cycle (a very simple calendar), so the week is one of the cycles of the Jewish calendar. It is also common to operate two calendars simultaneously, usually providing unrelated cycles, and the result may also be considered a more complex calendar. For example, the Gregorian calendar has no inherent dependence on the seven-day week, but in Western society the two are used together, and calendar tools indicate both the Gregorian date and the day of week.

Days used by solar calendars
Solar calendars assign a date to each solar day. A day may consist of the period between sunrise and sunset, with a following period of night, or it may be a period between successive events such as two sunsets. The length of the interval between two such successive events may be allowed to vary slightly during the year, or it may be averaged into a mean solar day. Other types of calendar may also use a solar day.

Future reform
There have been a number of proposals for reform of the calendar, such as the World calendar and International Fixed Calendar. The United Nations considered adopting such a reformed calendar for a while in the 1950s, but these proposals have lost most of their popularity. Holocene calendar is another one for counting years.

Lunar calendars

Not all calendars use the solar year as a unit. A lunar calendar is one in which days are numbered within each lunar phase cycle. Because the length of the lunar month is not an even fraction of the length of the tropical year, a purely lunar calendar quickly drifts against the seasons. It does, however, stay constant with respect to other phenomena, notably tides. A lunisolar calendar is a lunar calendar that compensates by adding an extra month as needed to realign the months with the seasons. An example is the Jewish calendar which uses a 19 year cycle.

Lunar calendars are believed to be the oldest calendars invented by mankind. Cro-Magnon people are claimed to have invented one around 32,000 BC.

Fiscal calendars

A fiscal calendar (such as a 5/4/4 calendar) fixes each month at a specific number of weeks to facilitate comparisons from month to month and year to year. January always has exactly 5 weeks (Sunday through Saturday), February has 4 weeks, March has 4 weeks, etc. Note that this calendar will normally need to add a 53rd week to every 5th or 6th year, which might be added to December or might not be, depending on how the organization uses those dates. There exists an international standard way to do this (the ISO week). The ISO week runs Monday through Sunday and Week 1 is always the week that contains January 4 Gregorian.

Calendar subdivisions
Nearly all calendar systems group consecutive days into "months" and also into "years". In a solar calendar a year approximates Earth's tropical year (that is, the time it takes for a complete cycle of seasons), traditionally used to facilitate the planning of agricultural activities. In a lunar calendar, the month approximates the cycle of the moon phase. Consecutive days may be grouped into other periods such as the week.

Because the number of days in the tropical year is not a whole number, a solar calendar must have a different number of days in different years. This may be handled, for example, by adding an extra day (29 February) in leap years. The same applies to months in a lunar calendar and also the number of months in a year in a lunisolar calendar. This is generally known as intercalation. Even if a calendar is solar, but not lunar, the year cannot be divided entirely into months that never vary in length.

Cultures may define other units of time, such as the week, for the purpose of scheduling regular activities that do not easily coincide with months or years.

Complete and incomplete calendars
Calendars may be either complete or incomplete. Complete calendars provide a way of naming each consecutive day, while incomplete calendars do not. The early Roman calendar, which had no way of designating the days of the winter months other than to lump them together as "winter", is an example of an incomplete calendar, while the Gregorian calendar is an example of a complete calendar.

A pragmatic calendar is based on observation; examples are the religious Islamic calendar and the old religious Jewish calendar in the time of the Second Temple. Such a calendar is also referred to as an observation-based or astronomical calendar. The advantage of such a calendar is that it is perfectly and perpetually accurate. The disadvantage is that working out when a particular date would occur is difficult.

A theoretical calendar is one that is based on a strict set of rules; an example is the current Jewish calendar. Such a calendar is also referred to a rule-based or arithmetical calendar. The advantage of such a calendar is the ease of working out when a particular date occurs. The disadvantage is imperfect accuracy. Furthermore if the calendar is very accurate, its accuracy perishes slowly over time owing to changes in Earth's rotation. This limits the lifetime of an accurate theoretical calendar to a few thousand years. After then, the rules would need to be modified from observations made since the invention of the calendar, resulting in a mixed calendar.

A mixed calendar combines the features of both pragmatic and theoretical calendars. Mixed calendars usually begin as theoretical calendars, but are adjusted pragmatically when some type of asynchrony becomes apparent; the shift from the Julian to the Gregorian calendar is such an example.

The Gregorian calendar, as a final example, is complete, solar, and mixed.

Calendars are also used as part of a complete timekeeping system: date and time of day together specify a moment in time. In the modern world, written calendars are no longer an essential part of such systems, as the advent of accurate clocks has made it possible to record time independently of astronomical events.

Also, the Persian calendar is used in Iran and Afghanistan. The Islamic calendar is used by Muslims the world over. The Chinese, Hebrew, Hindu, and Julian calendars are widely used for religious and/or social purposes.

Even where there is a commonly used calendar such as the Gregorian calendar, alternate calendars may also be used, such as a fiscal calendar.