If I knew the answer to this question, then I imagine I would be more famous than Stephen Hawking!
There are two ways of thinking about this. If you accept that the universe is infinite (that it goes on forever) then there is nothing outside space and instead the visible universe is just “stretching” expands. This isn’t very satisfying because its difficult for us to really understand infinity.
If, as you say, the univerise is finite (that it has a edge) then there must be something past that edge that space expanding into. The problem with this, is that because we are stuck within the bounds of space and the physics that exists inside it, there is no way we can observe anything outside it. So it is impossible for science to answer this question.
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