As you may all recall from your European history, Anarchist movements were popular on the Far Left in countries that had hereditary monarchies. The Anarchists believed that if the coercive apparatus of the monarchical state could be overthrown, people would spontaneously band together to do what is necessary to sustain society. In this respect, Anarchists were anti-authoritarian, Utopian, and communitarian. Like the communists with which they were often allied, they believed that the state would whither away over time.
In one very limited sense nearly all states that have abolished hereditary monarchies and aristocracies are “anarchist” states; however, in place of the old aristocracy we have an emergent corporate state, driven by the boom and bust cycles of capitalism and the opportunistic tendency of capitalists to consolidate their wealth and power during crisis in an almost inexorable march toward oligarchy.
Anarchists are all Libertarians today. That is to say, they pay lip service to minimalist government in exchange for certain rather limited “freedoms”;i.e., to do what they wish on what is left of their private property. But, I think they are being duped. While they are focused on “government” the corporate/capitalist class is gutting and privatizing government, creating a river of privatized wealth that flows to a more or less permanent aristocracy, and which only “trickles down” in under the heavy hand of their control over the levers of political power. For example, the masses may need and want something like health care, but corporate elites are able to block it because they are so rich and powerful that they are able to buy off Congress.
There are some communitarian societies, such as the Basques, who have made Anarchy work. You don’t hear much about them because their territory spans the borders of France, Spain and Portugal. They don’t need a state per se, because they have been granted a degree of separatist autonomy.
The Internet is also run on anarchist principles, and you can see the relentless assaults on it by corporations, working through the Bush Administration and the FCC, seeking to carve out some privatized advantage that will give them some non-competitive advantage. To the extent that we are all in favor of Net Neutrality, and resist those efforts, we are all Anarchists.