[masenghi]

Who do asteroids belong to? Can they just claim them and sell the material they extract?

Don't get me wrong, I would love to have my own asteroid mining company. Just need a couple billion dollars first. But what happens once there are two or more competing corporations? Who has the power to say which part of space belongs to whom? Will there be space battles?

[Symmetry]

Nobody owns or can own any celestial body according to the Outer Space Treaty[1]. Basically you can do whatever peaceful activity you want but you don't get to own anything in space except what you brought there. There is some legal question as to whether you own what you bring back, but you can certainly do whatever you want with it, including selling it.

Corporations operating in space are still subject to the laws of whatever national flag they're flying under, so I presume that the court system can come to some decision if more than one company wants to grab the same hunk of rock and metal. It'll probably be based on who launches their mission first, or something like that.