My two cents: No one is entirely sure about the "shape" of the universe right now. There are three possibilities, but they all illustrate the same general principle. 1) The universe is "closed." In this case, in principle, if you took a spaceship and flew far enough in one direction you'd end up back where you started. So the universe has no boundaries BUT it has a finite volume. An analogy might be to the surface of the earth--the surface has finite volume, but you can never find an "edge" to it while you're on it. The fact that the universe is "expanding" does NOT mean that the universe is taking over empty space. It means that the surface on which we live is getting larger. Think of it as if we were all two-dimensional beings that lived on the surface of a balloon. The balloon is the universe to us--we have no idea there's anything but the rubber of the balloon's skin. Then someone starts inflating the balloon. So we see it as the universe "expanding" in that there's more area for us to run around in. If I'm at one spot on the balloon, and someone else is on another spot, then as the balloon inflates we'll get further away from each other. But just because it's expanding doesn't mean that it has a boundary--it's still a sphere, and therefore has no edges. 2) The universe is "flat." In this case the universe would be infinite in a more traditional sense--you could go forever in one direction and you'd never come back to where you started. But even though it's infinite, it still expands. Using a balloon analogy again, say we had an infinite 2-dimensional sheet of rubber. It's total area is infinite, so obviously it can't "expand" in the traditional sense. But if I pull on it, it will stretch, and this will be viewed by the inhabitants of that sheet as expansion--again, if I'm stuck to one piece of rubber and my friend is stuck to another, we perceive ourselves as getting more distant from each other. And this process is occurring simultaneously at every point in the universe. 3) An "open" universe. This is shaped like a 4D hyperbola. It's very hard to visualize. The closest analogy is to a saddle but that really isn't a good enough analogy. The shape of the universe has implications for our ultimate fate. A closed universe will ultimately collapse back into itself (and maybe oscillate) and the universe will end in an enormous fireball. A flat universe will expand forever, but the expansion will slow down until the rate of expansion equals zero. An open universe will expand forever, but the rate of expansion will remain finite and nonzero. In flat and open universes the universe gets colder and colder forever. Open is considered the most likely possibility, and if you polled researchers you'd probably get 70% or more who picked this. (In fact, a fair amount of data suggests that the rate of expansion has been speeding up.) Flat is the next most likely. Closed is extremely unlikely--only a few diehards cling to this in the face of the data. The oscillating universe, philosophically appealing as it is, is basically dead.