That's a hard question. I won't even try to answer it myself. Here are some answers from Q&A websites I like:

"Perhaps the simplest way to look at these questions is the following: if the universe includes, by definition, everything -- all of space, time, matter, energy -- than there can be nothing outside of it (and hence no edge), nothing for it to expand into. Its true that this is contrary to our everyday experience, as is much else in physics and astronomy; but of course our everyday experience does not extend to the entire universe....Another way to look at it: if there were a higher-dimensional space in which the universe were embedded and into which it expands (like a two-dimensional balloon expanding into three-dimensional space), we could have no way of ever measuring the existence or characteristics of such a space. Whether such an unobservable space can truly be said to exist at all is a question best addressed by philosophers...!"

Reference: Ask a high-energy astronomer at NASA's Imagine the Universe website.

Here's Ned Wright's answer:

"This question is based on the ever popular misconception that the Universe is some curved object embedded in a higher dimensional space, and that the Universe is expanding into this space. This misconception is probably fostered by the balloon analogy which shows a 2-D spherical model of the Universe expanding in a 3-D space. While it is possible to think of the Universe this way, it is not necessary, and there is nothing whatsoever that we have measured or can measure that will show us anything about the larger space. Everything that we measure is within the Universe, and we see no edge or boundary or center of expansion. Thus the Universe is not expanding into anything that we can see, and this is not a profitable thing to think about. Just as Dali's Corpus Hypercubicus is just a 2-D picture of a 3-D object that represents the surface of a 4-D cube, remember that the balloon analogy is just a 2-D picture of a 3-D situation that is supposed to help you think about a curved 3-D space, but it does not mean that there is really a 4-D space that the Universe is expanding into."

Reference: Cosmology FAQ from UCLA astronomer's Ned Wright's excellent set of pages.