Last year NASA had a record breaking number of applicants to its astronaut training program. Over 18,000 people applied and only 12 made the cut. Of the 12, five are women!
According to NASA:
The first U.S. astronauts were selected in 1959, before the first human spaceflight operations even began. After stringent screening, NASA announced its selection of seven men, all pilots, as the first American astronauts. NASA has had 21 astronauts "classes" since the “Original Seven.” The backgrounds of NASA’s latest group of Astronaut Candidates include schoolteachers, doctors, scientists, and engineers.
If you think you have the right stuff, check out the application process on NASA's website.