Although we know the universe is expanding, nobody knows for sure what it is expanding into. Some scientists claim that it is not expanding into anything because nothing exists outside the Universe. Instead, space itself is stretching to accommodate the expanding matter. The Universe has no outside edge and no centre because the force of gravity distorts everything within it.
There is no edge to the universe, as far as we know. There’s an edge to the observable universe—we can only see so far out. That’s because light travels at a finite speed (one light-year per year), so as we look at distant things we’re also looking backward in time. Eventually we see what was happening almost 14 billion years ago, the remnant radiation from the Big Bang. That’s the Cosmic Microwave Background, which surrounds us from all sides. But it’s not really a physical “edge” in any useful sense.
Because we can only see so far, we’re not sure what things are like beyond our observable universe. The universe we do see is fairly uniform on large scales, and maybe that continues literally forever. Alternatively, the universe could wrap around like a (three-dimensional version of a) sphere or torus. If that were true, the universe would be finite in total size, but still wouldn’t have an edge, just like a circle doesn’t have a beginning or ending.
It’s also possible that the universe isn’t uniform past what we can see, and conditions are wildly different from place to place. That possibility is the cosmological multiverse. We don’t know if there is a multiverse in this sense, but since we can’t actually see one way or another, it’s wise to keep an open mind.
Okay, so we don’t actually think there is an edge to the universe. We think it either continues on infinitely far in all directions, or maybe it is wrapped up on itself so that it isn’t infinitely big, but still has no edges. The surface of a donut is like that: it doesn’t have an edge. It’s possible the whole universe is like that too (but in three dimensions—the surface of a donut is just two-dimensional). That means you could set off in any direction into space on a rocket ship, and if you traveled for long enough you would come back to where you started. No edges.
But there is also a thing we call the observable universe, which is the part of space that we can actually see. The edge of that is the place beyond which light hasn’t had time to reach us since the beginning of the universe. That’s only the edge of what we can see, and beyond that is probably more of the same stuff that we can see around us: super-clusters of galaxies, each enormous galaxy containing billions of stars and planets.
That depends on what you mean by the edge of the universe. Because the speed of light is finite, as we look farther and farther out in space, we look farther and farther back in time — even when we look at the galaxy next door, Andromeda, we see not what’s happening now, but what was happening two and a half millions of years ago when Andromeda’s stars emitted the light that our telescopes are only now detecting. The oldest light we can see has come from the farthest away, so in one sense, the edge of the universe is whatever we can see in the most ancient light that reaches us. In our universe, this is the cosmic microwave background — a faint, lingering afterglow of the Big Bang, marking when the universe cooled down enough to let atoms form. This is called the surface of last scattering, since it marks the place where photons stopped ping-ponging around between electrons in a hot, ionized plasma and started streaming out through transparent space, all the way across billions of light-years down to us on Earth. So you could say that the edge of the universe is the surface of last scattering.