I actually had to look all this up (and took legal advice from UK Government law officers) following complaints from concerned UK citizens (read: amateur astronomers). The advice on the BNSC website re naming of stars was drafted by me (although looking at it now, it seems to have reverted somewhat to its former incorrect position).<br /><br />Regarding Moon property, the Outer Space Treaty says: "Outer space, including the moon and other celestial bodies, is not subject to national appropriation by claim of sovereignty, by means of use or occupation, or by any other means." This means that you cannot establish a claim to land on the Moon within the legal systems of any of the signatory states, and that those states won't recognise any such claim under any other country's legal system either.<br /><br />However, there is nothing to stop any private organisation setting out a policy and asking people to abide by it, perhaps with a private arbitration service (a bit like the 'ownership' of domain names on the Internet, or the Court of Arbitration for Sport), although it's possible that such a private agreement could be expressed to comply with a particular law and be subject to a particular court's jurisdiction (like the Admiral's Cup). You'd probably need some sanctions against non-compliance - boycotts from transport and supply companies etc.<br /><br />Eventually, if there was a Moon colony that gained its independence and recognition as a state, it would not itself be bound by the Outer Space Treaty (unless the new state signed it, which seems unlikely), and it could enforce its own municipal law on the subject.