"Our objective is not to continue the Cold War, but to end it. We have reached an agreement at the United Nations on the peaceful uses of outer space," President Lyndon Johnson said.
In 1967, the United Nations laid down a set of rules to govern space exploration, and 104 nations signed on.
It's called the Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies. The Outer Space Treaty, for short. Here are the important bits.
Article I: Space exploration shall be the "province of all mankind," free of discrimination.
Article II: No nation can lay sovereign claim to any celestial territory. Space is for everyone.
Article III: Up here, international law and the U.N. Charter apply.
Article IV: No nukes, weapons of mass destruction or military maneuvers or bases.
Article V: Astronauts shall render all possible assistance to other astronauts.
Article VII: If you launch it, you're liable for any damage.
Article IX: No contaminating celestial bodies, and no contaminating Earth with anything you bring back.
Article XI: Parties shall disclose the nature, conduct, location and results of space activities.
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The relatively new field of commercial spaceflight has its own rules, typically managed by national agencies. In the U.S., the Federal Aviation Administration handles them.