NASA is developing a mission that will "identify, capture, and redirect" a chosen asteroid and set it in orbit around the moon, according to their website.
Two designs are being considered: one is a large inflatable bag-like system, the other is a robotic arm that would snag a sizable chunk off of a larger asteroid. Later this year, NASA will decide which approach to start with.
Astronauts aboard the Orion spacecraft will visit the captured asteroid in 2020 and will return with samples in order to study the asteroid's composition. As well as knowledge for knowledge's sake, this will provide NASA with an opportunity to test out equipment that can be used for a possible mars landing and will also prevent any asteroid/earth collisions, such as the one that killed the dinosaurs or the one featured in Michael Bay's movie Armageddon.
Asteroids are chunks of matter left over from the formation of the solar system, and studying them can lead to new insights on the formation of Earth. As well as looking back, asteroid study can also help us move forward. They can be literal stepping stones for space pioneers set to explore deeper into the solar system and also contain resources such as water that can be of use to astronauts traveling through space.
Since 2010, NASA has been keeping an eye on objects close to our planet via their Near Earth Object Human Space Flight Accessible Targets Study, and last year the Asteroid Initiative was launched which narrowed the focus specifically to asteroids. Since then, they've found about 1,217 Near Earth Asteroids (NEAs), ranging from the size of a car to larger ones the size of a small moon. Of those identified, six are seen as good candidates for the relocation project.