Territorial rights downwards

Anyone know if/ how far down countries can legally claim territorial rights below the earth’s suface.
Obviously a country claims the natural resources below it but what, legally, would be the situation if an adjacent country drilled at an angle to access and extract the resource.
If these rights exist, do they extend all the way to the Earth’s core?

I’m not sure it’s completely defined, but it’s worth noting that one of the causes of the first Gulf War was slant drilling (which is illegal by the various treaties making up international law) by Kuwait into Iraqi oil reserves (or at least it was one of the reasons given by Iraq - I don’t know the veracity of their claims as slant drilling is very difficult to trace).

The common law maxim was always “cuius est solum, eius est usque ad cœlum et ad infernos” i.e. if you own the soil then you own the air above it without limit and everything underneath it down to the earth’s core. In most places there are statutory limitations in place e.g. government ownership of mineral rights and control of airspace, but I suppose that it’s possible for one sovereign country to assert the principle against another sovereign country.

As a slight hijack, at what altitude does space begin? I know that a nation such as the USSR could rightly shoot down one of our U2 spy planes, but what is stopping them from shooting down a space shuttle or satellite?

The expense of doing so, plus the political aspects involved with shooting down something that belongs to the USA.

I don’t know how high the highest missiles fly, and I don’t think anyone who’s allowed to speak on the matter publicly knows. I do know, however, that `space’ is a somewhat amorphous concept when you get to suborbital craft: Did that experimental plane launch itself into space briefly, or did it propel itself into the outer reaches of our atmosphere? There was no air to speak of, but it didn’t achieve orbit. And so on.

This column may be of interest: Cecil Adams on Can I declare a “no-flight zone” over my house?