The most common theory is that of teh Expanding Universe. A Curved Universe would tend to imply a steady-state, non-changing Universe -- which goes against all of the observations we have seen up to this point. In fact, the "steady state" Universe has been proven to be impossible in 2003. [You can forget about the curved space stuff]
But remember, there are TWO parts to the Universe -- MATTER, and NOTHINGNESS.
MATTER is all the stuff IN the Universe -- stars and dust and galaxies and planets and moons and . . . . Nothingness is what is BETWEEN those "things". We call it "space". The combination of all the matter AND all the space between it is what we call ":OUTER SPACE"
Now here is a question for you: Where does a cloud "end"? What defines the edge of a cloud? it is CLEAR that clouds have edges (you can see them) but if you have ever been in an airplane you know that you can never FEEL the edge of a cloud. Therefore, the cloud "ends" when there is no more condensed water vapor present.
And so it is with the Universe. The "end of the Universe" is the point at which no more MATTER is present. If the Big Bang is treu, the matter is flying outwards, but at some point there is no more matter present, JUST NOTHINGNESS.
This nothingness is what the Universe is "expanding into". So, beyond the "edge of the Universe" is just more nothingness -- more space. Just like a clod, there is no hard "edge" or barrier at the End of the Universe. You just eventually reach a point where no more matter exists (except you).