Spacetime (probably) does indeed have at least one boundary. Crazy Buddy mentioned three related questions in his comment, and reading these will help you understand why spacetime has a boundary in the past i.e. the Big Bang. This is a singularity and it is a boundary because you cannot follow geodesics back through it to earlier times.
If the universe were closed (the experimental evidence is that it isn't) then there would be a similar boundary in the future i.e. the Big Crunch. This time it would be impossible to follow geodesics forward through the Big Crunch to later times.
In some of the theories of dark energy future boundaries may exist even though the universe is not closed. In particular there is a possible singularity called the Big Rip that would also act as a future boundary. However I should emphasise that these theories are highly speculative.
All the above boundaries are ones that co-moving observers hit by travelling in time. If you're asking if the universe has a boundary in space, i.e. there is a point where some observer could not move in that spatial direction, then as far as we know there are no such boundaries. If the universe is currently infinite then it has always been infinite. Alternatively if it is closed (presumably on some scale much large that the observable universe) then there would by definition be no edge.
On last caveat: I would guess most of us don't believe singularities exist, and some theory of quantum gravity will take over at very short distances and remove the singularity. This would also remove the boundaries I mentioned above. However no such theory of quantum gravity exists at the moment.
Response to comment:
It's a common misconception that the Big Bang was a point where everything came into existance, and therefore that the expanding universe must have an edge because it expanded from a point of finite size. However this is not the case. There are two possibilities:
- the universe is closed on some very large scale.
In this case consider the analogy of a balloon deflating. A (very small) ant crawling on the balloon would never encounter an edge as the balloon shrinks, so there would be no boundary. The Big Bang is analogous to the point where the ballon shrinks to zero size. What happens at zero size we can't say (because this is a singularity) but at any non-zero size, no matter how small, there is still no boundary.
- the universe is infinite
This is harder to stretch your brain around. If the universe is infinite then it has always been infinite and remains infinite in size even back to the Big Bang. At the Big bang you get the odd result that the spacing between every point in the universe is zero, but the universe is still infinite. But then this is what makes the Big Bang a singularity. Whatever the case, like the closed universe for any non-zero size, no matter how small, there is still no boundary because the universe is infinite.