space - What happens outside of the universe?

There are a number of ways of answering this question. Here are a few.

Firstly we can simply state that, by definition, the universe is everything and therefore there is simply nothing beyond it. Not nothing as in empty space, but nothing at all, not even space (or it could be that the universe is infinite.)

Secondly, we could look at what we mean when we say that the universe is expanding. What this really means is that the space-time metric is growing. It does not mean that the universe is expanding into some void, but that distances inside the universe are increasing with time. My favourite parallel is with an balloon being blown up. Points on the surface get further away without the addition of any more material to the balloon. (It is, of course, an imperfect parallel as the balloon is expanding in a medium).

Thirdly, and now we are touching on metaphysics (in the sense that it may be impossible to test some or any of these theories), we can say that "our" universe is one of many and that, actually it may well have an edge which even now is crashing into some other universe, or perhaps we are inside a black hole inside another universe. These multiverse theories are ten-a-penny but they are not without scientific merit, possibly.