It's an interesting question Joe, and the answer depends on how you look at it.
Space seems to be unbounded, which is to say that we're not aware of any intrinsic limits to it, so in this respect it appears to be infinite, or at least potentially so. However, if you go along with the idea of the Big Bang, then the universe clearly appears to be finite in the time dimension, for it has a finite age.
The thing is though, that we cannot really separate space and time from each other and need to think of them as a unified space-time environment, which is expanding both in space and in time: the galaxies all seem to be receding from each other due to the spatial expansion of the universe, and at the same time we also seem to be constantly moving into the future i.e. into new time.
When thinking about space-time though, you need to be aware of the limitations of having a view point that is restricted to a particular number of dimensions, for there is infinite n-1 dimensional space within a finite n-dimensional environment e.g. there is enough space for an infinitely long 1-dimensional line within a finite 2-dimensional area. This implies then, that there would seem to be scope for infinite 3-dimensional space-time within our 4-dimensional space-time environment, which is currently finite in the time dimension. However, while the universe appears to currently be finite in the time dimension, the time dimension appears to be unbounded, just like space, which in turn rather suggests that our 4-dimensional space-time may exist within a higher order 5-dimensional space-time environment.