It is a question that is becoming more relevant as businesses and other entities consider "space mining," or excavating minerals and other potentially precious materials from the moon, meteors, and other intergalactic celestial bodies.

It's an emerging area of the law because, unlike the shores of the ocean which are considered to be a part of the country within certain number of miles, bodies in space are by definition not a part of previously governed land.

The laws of "finding" generally vary according to where you find something. For example, if you come across a broach which is inside of a building, it is considered to be a part of the property upon which you found it. Alternatively, if you find an article of clothing in a public place, then it is only properly yours if the person abandoned it--i.e. they did not intend to return to the place of loss to reclaim the property. Buried treasure, alternatively, is considered to belong to the person who owns the land within which it is contained, unless it is within unclaimed territory, in which case it belongs to whomever excavates it. See for example Legal Battle over Spanish Treasure.

The current theory regarding ownership over space property is that whomever properly claims and excavates it is entitled to the property rights. Because there is no central governmental authority regulating ownership over space properties, there is no one to prevent prospectors from making gains.

The situation regarding the moon in particular strikes me as identical to that of Antarctica. In the case of Antarctica, a treaty was signed whereby the agreeing parties decided that individuals shall be governed by the laws of their individual countries, and disputes are to be handled by an International Court of Justice. It shall be interesting to see if space becomes (in the legal sense) the next South Pole.