Officially, nobody. The issue of owning extraterrestrial space or objects is governed by the United Nations’ Outer Space Treaty, first enacted in 1967. It forms the basis of international space law. Article II of the treaty states “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.”
Some entrepreneurs have argued that this doesn’t apply to private individuals and they are therefore free to claim (and sell off) portions of outer space, including lunar real-estate! However, legal bigwigs insist that nations take responsibility for not only state and corporate functions but also private citizens. There is also a legal concept called corpus possessionis which implies that for something to be ‘owned’ it must be ‘controlled’ – something not possible unless an individual can access and defend their lunar real-estate. In reality, anyone buying a piece of the Moon owns nothing more than an expensive bit of paper.