Apparently no one.
The Outer Space Treaty provides a legal framework for the international use of space for peaceful purposes, including the moon and other celestial bodies. Widely considered the "Magna Carta of space law," this treaty lays down the fundamental principle of non-appropriation and that the exploration and use of space shall be the province of all mankind.
According to the treaty, states bear international responsibility for national activities in space, including by non-governmental entities. The Outer Space Treaty says governments cannot claim ownership of the lunar surface and that stations and installations on the moon shall be open to others, said Hedman.
Who owns the moon? It's 'complicated,' say experts - CNN.com