We will may never have a way of knowing what lies outside of the known universe, or if there is an "outside." This is a question that will be asked as long as humans exist, and asked again by what ever intelligent lifeforms replace us.
We can see no boundaries, but it's doubtful that it's infinite. It had a beginning, and it's expansion rate is per volume is known, so it's probably just very large with it's boundaries out of reach to our vision. It's expansion rate would have had to have been at an infinite speed for it to be infinite in size and yet have a certain age. Being as we can measure a portion of it's rate of expansion, and it's rate is accelerating, that means that it's speed is not infinite, so it's size then is not infinite.
According to one theory, it doesn't really end, but it does circle back on itself. In other words, if you left earth and traveled a straight line, you would eventually end up at the same point that you started from. Sort of if all of space existed on the outside of a globe. If you start from any point on the globe, travel straight in one direction you will come back to the same point.
If I had a staff, and I went to the edge of the universe and stuck my staff past the edge, that would become the new end of the universe. Also, there is the possibility of a multiverse, so if we went out of the universe we could go into other universes. Furthermore, the universe is always expanding since the Big Bang, unless it stops and either stays at that point or collapses in on itself, this is called the Big Crunch.