I think of the universe as finite but unbounded. Sort of like the two-dimensional surface of a sphere, which curves into a third dimension.
If you restrict yourself to that 2-D surface like a man either walking or swimming on the earth's surface, (without flying or digging), you can travel forever without coming to an edge or boundary, although you may occasionally approach a spot you once departed from the opposite direction you left it! The surface of a sphere has neither boundary nor center, but it does not go on forever. Remember, in this analogy, leaving the earth's surface is not allowed, just as leaving 3-D spacetime is not allowed in our own space.
The universe is also finite in space (since it is finite in time, it hasn't had time to become infinitely big) but it has no edge or center. It is a three dimensional volume which curves into a fourth dimension (time?).
Travel in a straight line any direction and in a finite amount of time you will come back to where you started from, providing you can move fast enough and the universe expands slowly enough.
Now let us speculate: since the speed of light provides us an upper speed limit, there may be parts of the expanding universe we can never visit, because the universe will not be old enough for enough time to have passed for us to get there. A place that can never be reached can be thought of as a boundary, or edge, but that is only because the universe is expanding and our own speed is finite. Whether or not such a "receding boundary" exists I don't think anyone really knows.
Due to Hubble's Law, the observed velocity of recession of distant parts of the universe increases the further out we look, right now we have detected distant objects receding from us at a substantial fraction of the speed of light. Presumably, when the velocity of recession approaches the speed of light, that's where the edge is. But if we aimed a light beam towards that edge, by the time it got there the edge would have receded further, and be moving even faster, our light beam might never get there, no matter how much time it had. We can't even calculate possibilities for this because we don't know the precise value of the Hubble Constant (how recession velocity increases as a function of distance), or even it it's linear or asymptotic to c at great distances. All we do know is the furthest objects we can see (distant quasars) are the light signatures of events that ocurred when the universe was about one quarter its current age. This means that they are much further away now, it's taken so long for their light to get here.
Still we know we can actually see the Big Bang, the flash from the explosion may have originated in the center of the young universe, but now we see it in every direction we look, the entire sky glows feebly with its faint whisper, the 3 degree Kelvin blackbody radiation. It is the blinding gamma-ray flash of the Big Bang detonation itself, red-shifted down into a dim microwave hiss from the very edge of eternity.