Does US own the Moon?

"Nearly 40 years after the U.S. flag was planted on the moon, a global rush to the final frontier has some pondering property rights out there.

India, Japan and China are now circling the moon with their respective spacecraft – to be joined next year by NASA's Lunar Reconnaissance Orbiter. Then there's the Google Lunar X Prize, a $30 million competition for the first privately funded team to send a robot to the moon, travel some 1,640 feet (500 meters) and transmit video, images and data back to Earth."

This could lead to a new frontier of Real Estate.
Looking at the mess US did of Earth Real Estate I think it would be rather unwise to do the same there....