The Space Age is generally considered to have begun October 4, 1957, when the Soviet Union launched the first man-made item into orbit around the Earth, Sputnik (which translates literally as "Satellite").
The Space Age will presumably never end, as it involves not just manned missions into space, but the many unmanned probes launched into space to explore the farthest reaches of the known universe. Currently, there are probes on the surface of Mars, orbiting Saturn and its rings and moons, orbiting the Sun, and so on. One of the earlier series of such unmanned probes are Voyager I and II, both launched in 1977. Expected to have failed or crashed by now, the Voyager spacecraft instead are still going strong, and Voyager I is one the verge of exiting our Solar System for points unknown. In the meantime, NASA and its European and Russian counterparts continue to design and launch more probes for scientific research.
The Space Age is not something that has an exact definition and exact dates. Most people would likely say that we are still in the Space Age.
One way to define the Space Age is that it was the time in which space exploration was important in the world. This clearly started in 1957 with Sputnik. It can be said to have peaked with the manned moon missions launched by the United States. These ended in 1972, so it would be possible to argue that the Space Age ended then.
However, this does not seem plausible as space exploration has continued to be important. China put a man in space in 2003. The US space shuttle program continued until 2011. There is still plenty of interest in various kinds of space exploration. Therefore, I think that it is best to say that the Space Age has not yet come to an end.