The Outer Space Treaty is the relevant law here and it makes it clear, that no country can claim territory anywhere outside of the Earth. All you can claim as your own is your habitat that you made. And - property rights based on territory such as we have here on the Earth - they are inconsistent with the treaty.
However - that doesn't matter as much as you might think. After all the land area in space is pretty much worthless unless you build habitats on it. You might be able to grow crops on the Moon in greenhouses, but there is no way you can do anything much with the bare landscape. And your greenhouses would be owned by you.
And - we don't want countries to be able to lay claim to the landscape itself - for military bases or some such, because the aim of the treaty is peaceful exploration so that is explicitly prohibited.
So - there are several ideas about how this could be taken further - the OST permits anyone to build habitats in space, so long as it is consistent with planetary protection and not interfering with the legitimate scientific and other interests of other parties to the treaty. But it doesn't protect those habitats. For instance, nothing to say that someone else can't come along and put a habitat right next to your one, or go inside it when you aren't there - or indeed - to just take it apart and return it to Earth - the only thing it says is that if you return it to Earth you have to give it back to the owners of the habitat.
So - probably the treaty needs to be extended in various ways. One idea is to do with the way that an oil platform has a zone around it where the owners of the platform are in control - they can tell ships whether it is okay to approach and are in command of visiting ships as they approach - for safety reasons. Same for the ISS - it is in command of visiting spaceships when they get close to the ISS again for safety reasons.
So then the idea is that as for the ISS, if you own a habitat in space - then you are in command of visiting spacecraft as soon as they get within some proximity zone, and also similarly in command of an area immediately around your habitat on the surface. But this is just an idea - nobody has attempted to put it into law yet.
Another idea is to do with functional ownership - if you start a mine in space for instance - say mining minerals and metals from an asteroid - that so long as you keep it functioning - keep a presence there - that you have a functional ownership of that mine and others can't take it over from you - though you don't own the resources you mine. But if you abandon the mine, others would be able to move in and take it over.
There are other ideas also - I'm not an expert here but there are lawyers who are expert in space law - and some of them have elaborated fairly detailed complex ideas about how this might be dealt with in future developments.
As for establishing colonies on Mars, personally I think that is unlikely to happen in the near future, even if we get the technological ability to send humans there at a reasonable cost - see some of my other answers for the reasons.
At any rate in the near future, then if anyone wants to send humans to Mars they have to comply with the OST and with planetary protection requirements.
And longer term in the future - well the Outer Space Treaty is something very precious. It is extremely hard to get countries to agree to a space treaty and there have been failed treaties such as the Moon treaty.
The OST is the main thing that preserves peace in space. Indeed before they signed it, the US had already drawn up some reasonably detailed plans for a military colony on the Moon. They shelved those plans after they signed the OST.
So, probably, without it, the Moon would have military colonies from the US, USSR and China and probably other countries as well and we'd have weapons of mass destruction stationed on the Moon also by now, and in orbit around the Earth (the OST explicitly prohibits stationing weapons of mass destruction in orbit or anywhere in space outside of the Earth) - and I think that would probably have been a more dangerous world to live in politically than the world we are in now.
So - I think that if at all possible, that even space colonies, if we get them - that they will want to retain the OST. And probably more regulations also. Because a space colony would be tremendously fragile. Even a hit by a spaceship that fails to retro fire its engines and stop cleanly would destroy the station utterly. Never mind hostile action.
Also - it's not likely that we can modify the Outer Space Treaty at this stage. Its signatories include all the space faring nations, all the aspiring space faring nations (including N. Korea), and just about all the other nations as well. It's not at all easy to get those diverse nation states of widely differing political systems and ideologies and cultures to agree on anything, so the OST is a remarkable achievement.
So any future laws in space are likely to be additions to the space treaty rather than attempts to modify it. And not likely that, e..g the US would withdraw from the OST because the political implications of that would be vast and uncertain. For instance that other nations might then feel that they could and should place weapons of mass destruction in orbit, or be concerned that the US was planning to do this.
Sometimes people in internet forum debates and such-like suggest that the US should withdraw from the OST in order to permit ownership in space, but I've never heard anyone suggest that seriously. And I think myself that surely we need to work within it - the benefits of doing that far outweigh any benefits of withdrawing from it.
Individuals however - they often make claims of ownership - of the sun, of the Moon, of the galaxy, sometimes just for fun or to make a legal point - but there is no government able to back up these claims as it is not permitted to do so under the OST. So such claims are meaningless as they can't be enforced by anyone.
Incidentally the OST also explicitly says that the governments are responsible for the activities of its citizens in space (so that is irrespective of how they get there) - and that in the case of an international organization, that all the governments of the citizens involved in that organization are responsible for making sure it upholds the OST. So it's not possible for an organization to claim territory in space either.
So we won't get a territorial claim in space by SpaceX or Virgin Galactica or such like. At least - the relevant governments (UK, US etc) would be required under the OST to repudiate any such claim.
And unlike the Antarctica treaty - the OST is for all future time, with no review date or time out.
There is a legal quibble some have suggested here - if you move an asteroid, does it then no longer become a space object in the sense of the OST - but rather an artificial object like a habitat? Is the law limited only to objects too large for us to move them? Might that be a way to assert ownership of small asteroids and later larger astronomical objects consistently with the OST - by moving them?
I don't think many accept this idea as valid though.
And BTW that also leads to an issue with space mining. According to the OST we go into space for the benefit of all humanity. So it is not clear what is the legal status of materials returned to the Earth from space by a space mining company. Do they have the right to sell this to the rest of us? Or can we claim that e.g., if they go into space and locate a platinum asteroid and start mining it to return it to Earth - that - since we are exploring space for the benefit of humanity - they are just returning what is already ours? Or at least, is ours as much as it is anyone's?
On what basis do they have that right to sell this asteroid to us? This is a gray area of the law that would need to be clarified, and it is not covered in the OST. Some have suggested that we need to build in some kind of method of "benefit sharing" into space law to make sure that all nations of the Earth benefit from return of resources to the Earth.Or some other way to make sure that space exploration continues to be carried out in a manner of benefit to all of humanity.
See also:
Robert Walker's answer to Is it lawful or do laws even exist in the instance of a private space company like SpaceX claiming a part of a celestial body as its territory?
Robert Walker's answer to How feasible is Elon Musk's idea to establish a colony on Mars in the 2020s?
However - that doesn't matter as much as you might think. After all the land area in space is pretty much worthless unless you build habitats on it. You might be able to grow crops on the Moon in greenhouses, but there is no way you can do anything much with the bare landscape. And your greenhouses would be owned by you.
And - we don't want countries to be able to lay claim to the landscape itself - for military bases or some such, because the aim of the treaty is peaceful exploration so that is explicitly prohibited.
So - there are several ideas about how this could be taken further - the OST permits anyone to build habitats in space, so long as it is consistent with planetary protection and not interfering with the legitimate scientific and other interests of other parties to the treaty. But it doesn't protect those habitats. For instance, nothing to say that someone else can't come along and put a habitat right next to your one, or go inside it when you aren't there - or indeed - to just take it apart and return it to Earth - the only thing it says is that if you return it to Earth you have to give it back to the owners of the habitat.
So - probably the treaty needs to be extended in various ways. One idea is to do with the way that an oil platform has a zone around it where the owners of the platform are in control - they can tell ships whether it is okay to approach and are in command of visiting ships as they approach - for safety reasons. Same for the ISS - it is in command of visiting spaceships when they get close to the ISS again for safety reasons.
So then the idea is that as for the ISS, if you own a habitat in space - then you are in command of visiting spacecraft as soon as they get within some proximity zone, and also similarly in command of an area immediately around your habitat on the surface. But this is just an idea - nobody has attempted to put it into law yet.
Another idea is to do with functional ownership - if you start a mine in space for instance - say mining minerals and metals from an asteroid - that so long as you keep it functioning - keep a presence there - that you have a functional ownership of that mine and others can't take it over from you - though you don't own the resources you mine. But if you abandon the mine, others would be able to move in and take it over.
There are other ideas also - I'm not an expert here but there are lawyers who are expert in space law - and some of them have elaborated fairly detailed complex ideas about how this might be dealt with in future developments.
As for establishing colonies on Mars, personally I think that is unlikely to happen in the near future, even if we get the technological ability to send humans there at a reasonable cost - see some of my other answers for the reasons.
At any rate in the near future, then if anyone wants to send humans to Mars they have to comply with the OST and with planetary protection requirements.
And longer term in the future - well the Outer Space Treaty is something very precious. It is extremely hard to get countries to agree to a space treaty and there have been failed treaties such as the Moon treaty.
The OST is the main thing that preserves peace in space. Indeed before they signed it, the US had already drawn up some reasonably detailed plans for a military colony on the Moon. They shelved those plans after they signed the OST.
So, probably, without it, the Moon would have military colonies from the US, USSR and China and probably other countries as well and we'd have weapons of mass destruction stationed on the Moon also by now, and in orbit around the Earth (the OST explicitly prohibits stationing weapons of mass destruction in orbit or anywhere in space outside of the Earth) - and I think that would probably have been a more dangerous world to live in politically than the world we are in now.
So - I think that if at all possible, that even space colonies, if we get them - that they will want to retain the OST. And probably more regulations also. Because a space colony would be tremendously fragile. Even a hit by a spaceship that fails to retro fire its engines and stop cleanly would destroy the station utterly. Never mind hostile action.
Also - it's not likely that we can modify the Outer Space Treaty at this stage. Its signatories include all the space faring nations, all the aspiring space faring nations (including N. Korea), and just about all the other nations as well. It's not at all easy to get those diverse nation states of widely differing political systems and ideologies and cultures to agree on anything, so the OST is a remarkable achievement.
So any future laws in space are likely to be additions to the space treaty rather than attempts to modify it. And not likely that, e..g the US would withdraw from the OST because the political implications of that would be vast and uncertain. For instance that other nations might then feel that they could and should place weapons of mass destruction in orbit, or be concerned that the US was planning to do this.
Sometimes people in internet forum debates and such-like suggest that the US should withdraw from the OST in order to permit ownership in space, but I've never heard anyone suggest that seriously. And I think myself that surely we need to work within it - the benefits of doing that far outweigh any benefits of withdrawing from it.
Individuals however - they often make claims of ownership - of the sun, of the Moon, of the galaxy, sometimes just for fun or to make a legal point - but there is no government able to back up these claims as it is not permitted to do so under the OST. So such claims are meaningless as they can't be enforced by anyone.
Incidentally the OST also explicitly says that the governments are responsible for the activities of its citizens in space (so that is irrespective of how they get there) - and that in the case of an international organization, that all the governments of the citizens involved in that organization are responsible for making sure it upholds the OST. So it's not possible for an organization to claim territory in space either.
So we won't get a territorial claim in space by SpaceX or Virgin Galactica or such like. At least - the relevant governments (UK, US etc) would be required under the OST to repudiate any such claim.
And unlike the Antarctica treaty - the OST is for all future time, with no review date or time out.
There is a legal quibble some have suggested here - if you move an asteroid, does it then no longer become a space object in the sense of the OST - but rather an artificial object like a habitat? Is the law limited only to objects too large for us to move them? Might that be a way to assert ownership of small asteroids and later larger astronomical objects consistently with the OST - by moving them?
I don't think many accept this idea as valid though.
And BTW that also leads to an issue with space mining. According to the OST we go into space for the benefit of all humanity. So it is not clear what is the legal status of materials returned to the Earth from space by a space mining company. Do they have the right to sell this to the rest of us? Or can we claim that e.g., if they go into space and locate a platinum asteroid and start mining it to return it to Earth - that - since we are exploring space for the benefit of humanity - they are just returning what is already ours? Or at least, is ours as much as it is anyone's?
On what basis do they have that right to sell this asteroid to us? This is a gray area of the law that would need to be clarified, and it is not covered in the OST. Some have suggested that we need to build in some kind of method of "benefit sharing" into space law to make sure that all nations of the Earth benefit from return of resources to the Earth.Or some other way to make sure that space exploration continues to be carried out in a manner of benefit to all of humanity.
See also:
Robert Walker's answer to Is it lawful or do laws even exist in the instance of a private space company like SpaceX claiming a part of a celestial body as its territory?
Robert Walker's answer to How feasible is Elon Musk's idea to establish a colony on Mars in the 2020s?
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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