As of 2014, NASA reports that astrophysicists are not yet aware of whether space ends or not. The governmental agency reports that although there is a finite limit to the amount of space visible from earth, this is merely because any light further than 15 to 20 billion light years from earth has not yet reached the planet.
New Scientist tackles the question by explaining that although infinity is mathematically conceptual, an infinite universe would not necessarily mean infinite space. The website explains that in the same way one could technically traverse the earth forever, there would still be finite space on which to travel. From Quarks to Quasars explains that due to the theories of special relativity and cosmic inflation, the universe should be at least significantly larger than the edge of the known universe. The website explains the popular theory that the universe is constantly expanding, so there may be no precise end to outer space at all. For practical purposes, the edge of the known universe can be considered the edge of space since it is currently impossible to see past anything more than 20 billion light years away, but theories from String Theory to Dark Flow speculate on the endless possibilities that may exist beyond that light barrier.