It may sound bizarre, but it's interesting to note that many of the world's top intellectuals are anarchists. These include Noam Chomsky and Howard Zinn (see here). This has prompted me to look into it a tiny bit. I watched a series of YouTube clips called Anarchism 101 that is here. It introduces the concept and addresses common objections. Seems very reasonable to me. There are different strands, including anarcho capitalism and anarcho syndicalism. The former is regarded as a contradiction in terms by it's critics (see here) and is criticized in the Anarchism 101 series. Chomsky and Zinn regard themselves as anarcho syndicalists.
Which societies have adopted anarchism? According to my first wiki link some Native American societies would qualify, if imperfectly. David Friedman asks whether the Amish are anarchists here and concludes that they are. For a brief period Spain was anarchic (see here). Monty Python's annoying peasant was an anarcho syndicalist (see here). A city in Hong Kong was regarded by some as anarcho capitalist and there's a German documentary on it starting here. It has since been demolished. Noam Chomsky criticizes anarcho capitalism and discusses liberterianism here.