Another interesting idea is the proposal to introduce a law of "Space Salvage". At sea the long-standing law of salvage allows the person who takes control of an abandoned vessel to claim ownership. One of the growing problems in Earth orbit is the amount of "space debris" - abandoned satellites, rocket stages and other pieces abandoned by the governments which launched them. By introducing a law of salvage there would be a strong incentive for businesses to collect together useful objects. Because of the high cost of launch, any mass in orbit is valuable. Even at a launch cost as low as $100/kg, scrap metal would be worth at least $100,000/ton in low orbit! And so we can foresee that recycling is sure to become a major orbital business.
When mining companies start operating on the Moon they will be under the auspices of the 1966 "Outer Space Treaty", that is the UN Treaty on the Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies (pretty sweeping!). The 1979 "Moon Agreement", that is, The Agreement Governing the Activities of States on the Moon and Other Celestial Bodie discusses this matter in more detail, but has been ratified by only a small number of countries, since most countries consider that it does not provide a good legal framework for business activity on the Moon.
We should remember that a certain amount of fundamental law is made "de facto", that is by someone doing something, and thereby establishing the fundamental principle that they may do it. A famous case of this was the flight of Sputnik 1 in 1957 which established the principle that spacecraft in orbit may overfly any nation - which is not allowed within the atmosphere without permission.
We may well see companies starting commercial operations on the lunar surface, perhaps Lunacorp and Celestis thereby proving that they may. And at a later date a company may have to defend its operations by force against "pirates", thereby establishing the right to self-defense - one of the fundamental rights of pioneers in new territory!
Much law that is required will be basically a simple extension of aviation law. Aviation is a huge, popular, profitable, global business, operating within a network of international law. It will be much simpler to add to this to cover rocket vehicles and flights to and from Earth orbit, than to start from scratch. Indeed the fact that the Office of Commercial Space Transportation was moved into the FAA in 1995, and the X-33 test-vehicle is to be licensed as an experimental aircraft are signs of this.