As others have noted, there are a couple of international treaties that say that no nation owns or can own the Moon. Whether anyone will care about those treaties by the time technology reaches the point where it would actually be possible for anyone to live on the Moon or make use of it in any realistic way is a very different question.
It wasn't until I read PearsonArtPhoto's post here that I learned that none of the countries who have actually launched spacecraft have ratified the treaty. (Hey, I learned something today!) So that makes it pretty irrelevant. But even if tomorrow the U.S., Russia, China, India, Japan, whomever, all ratified it, those nations may not even exist by the time someone wants to build a lunar colony, and if they do, they might simply renounce the treaty as out-dated.
Like, does anyone today pay any attention whatsoever to the Treaty of Tordesillas? Back in 1494, Spain and Portugal signed a treaty dividing the New World between them. Does anyone imagine that that treating is binding on the United States and Canada today?
Realistically, when technology reaches the point that a lunar colony is possible, I think someone will just plant the colony and declare that they own that little piece of the Moon. Then others will follow and divide it up. If we're lucky, the Moon will end up being carved up among various nations on a first-come/first-served basis. More likely, if it proves valuable enough economically or politically, wars will be fought to establish the borders. Barring some sudden and dramatic change in human nature unprecedented in history.