Technically part of Australia - an "external territory" or something along those lines. They claim a fair degree of independence, so we always have to fly from the international airport and take a passport or some other kind of ID to get in, which may actually have a bit more to do with the fact that they have more restrictive customs regulations than the mainland does.
On the good side, their independence means that they don't charge most of the taxes we get slugged on the mainland. Most of the GST sorts of things are simply non-existent, so there's always a list of things that we go shopping for when we visit.