Who owns it?? Nobody, .... and everybody. It seems the "law" concerning the moon is a bit complicated. The US has plans to go back and establish a base. Other countries have plans as well. When we drew up the Moon Treaty there were only two space faring nations, the US and Russia. But now there are more. And sooner than later these nations want to go to the moon. And do what?? I don't know.
We also have people that say the own the moon and so forth selling one acre plots of land. ASnd by the looks of it, people have made some money in the process. You can buy moon property right now at a cost of $18.95 to $37.50 an acre depending where you are. Check it out here.
But now I see this: http://moonpublicity.com/mp/
If I look up someday and see a Walmart sign in the half moon phase, I'll know that human beings have gone completely mad. Are we really about to scrape the moon in an effort of global marketing. Sigh....