The United Nations Office for Outer Space Affairs was formed in December of 1958. This little-talked-about group is responsible for implementing decisions of the United Nations General Assembly and the Committee on the Peaceful Uses of Outer Space. If you were ever wanting to practice international space law, this is where you need to be. They have made five treaties and agreements since their inception.
So, in summary, NO ONE owns space. Now when someone tries to sell you a plot on the moon or your name on a star, ask yourself how someone can sell something they don't own. Nuclear weapons are NOT allowed in space. (Nuclear power is, however, and is regulated as one of the five declarations and legal principles.) Keep track of your space junk (a growing issue), and if one of your satellites falls back to Earth, be prepared to pay for the consequences. If an astronaut makes an emergency landing in a country other than that which it launched from, it is due rescue services and to be returned to the country of origin. The fifth point becomes interesting as more plans are made to head out to solar system bodies, such as asteroids, and mine them for material resources, along with the planned base on the Moon and robotic explorations of other planets.