In calendars based on the Christian Era or Common Era, such as the Gregorian calendar, the 21st century is the current century, as of this writing, lasting from 2001-2100. The 21st century is the first century of the 3rd millennium (2001 - 3000). There was no year '0', so the first century began in year 1. See also . Despite this, common usage sometimes incorrectly regards the 21st century as lasting from 2000 to 2099. Decades are almost always considered as starting with the "0" year and named accordingly ("2010s", etc.), so the first decade of a century technically overlaps back into the preceding one.
Similar to the 20th century's place in popular culture as part of names such as 20th Century Fox, the 21st century has been used in the names of a number of companies and organizations. The real estate firm Century 21 uses a name that refers to this century.
The 21st century has had an influence on culture since well before it began. Speculation about future social, cultural, and technological trends frequently centered on the year 2000, starting with late-19th century essays and novels (often of a utopian nature) such as Edward Bellamy's Looking Backward. It's been said that the event horizon of Western culture was steadily shrinking in this period, since as late as the 1990s people were still often focusing on the year 2000 in their discussions of the future.
Religious beliefs in a "millennial apocalypse" were supplemented by genuine concerns about the Y2K computer "bug" and about possible terrorist attacks centered on the year-2000 celebrations, but the actual turn of the millennium (both the popularly-celebrated one in 2000 and the "purist" one in 2001) went by in a fairly anticlimactic manner.
However, the years since have continued in the tumultuous manner people of the 20th century were accustomed to expect, with wars, terrorism, and other conflict, as well as continued advances in science and technology including the continuing expansion of the use of computers and the Internet (despite the "tech bubble burst" where the overexuberance of early Internet companies was deflated).
More Y2K-style computer date failures are due before the end of the 21st century; the Unix datestamps, consisting of a count of the number of seconds since 1970, will overflow in 2038, while the family of operating systems descended from MS-DOS (including the various versions of Microsoft Windows) can't handle dates beyond 2099.
2005 UN Security Council decides war criminals in Darfur will be tried by the International Criminal Court (Resolution 1593) 
Science and technology
2001Apple Computer introduces iPod; portable digital music players become wildly popular and addresses raising music industry concerns about piracy with an integrated platform and Apple's Digital Rights Management format.
Some of the things that have dominated discussion and debate in this century include:
Overpopulation. The rapid growth of the world population results in more strain on ecosystems and more consumption of limited resources. Profligate population growth is a root cause for the proliferation of disease and scarcity-related problems such as poverty, famine, and economic instability. The United Nations estimates that world population will reach 9.1 billion by mid-century. The results of population growth seen in the last century are likely to be even more pronounced in this one. The burden placed on the planet's natural systems by humans is becoming unsustainable. Historically, large outbreaks of violence often coincide with a large increase in population density.
Poverty. Poverty remains the root cause of many of the world's other ills, including famine, disease, and insufficient education. Poverty contains many self-reinforcing elements (for instance, poverty can make education an unaffordable luxury, which tends to result in continuing poverty) that various aid groups hope to rectify in this century.
Climate change. Some scientists believe that a climate disaster is approaching if humanity does not change its pattern of environmental discharges. Others dispute this. Trends such as global warming, pollution, biodiversity loss and resource depletion all are growing factors that will contribute to significant issues in this century. Water in particular is an area of serious concern. Another instance of significant resource depletion is evident in oil production, which some scholars predict will reach a peak in this century, then begin a permanent downward trend.
Intellectual property. The increasing popularity of digital formats for entertainment media such as movies and music, and the ease of copying and distributing it via the Internet and peer-to-peer networks, has raised concerns in the media industry about piracy. Much debate is proceeding about the proper bounds between protection of copyright, trademark and patent rights versus fair use and the public domain, where some argue that such laws have shifted greatly towards intellectual property owners and away from the interests of the general public in recent years, while others say that such legal change is needed to deal with the threat of new technologies against the rights of authors and artists (or, as others put it, against the outmoded business models of the current entertainment industry). Domain name "cybersquatting" and access to patented drugs to combat epidemics in third-world countries are other IP concerns.
October 1, 2044 - Occultation of Regulus by Venus. The last was on July 7th, 1959. After 2044 the next occultation of Regulus by Venus will occur on October 21st, 3187, although some sources claim it will occur again on October 6th, 2271.
November 22, 2065 12:45 UTC - Venus will occult Jupiter. This event will be the first occultation of a planet by another since January 3, 1818. Unfortunately this event will be very difficult to observe, because the elongation of Venus and Jupiter from the sun on November 22, 2065 will be only 7 degrees.
2066: Triple conjunction Jupiter-Uranus
July 15, 2067 11:56 UTC - Mercury will occult Neptune. Unfortunately this rare event will be very difficult to observe.