Is it possible to destroy the asteroid in space itself? If it is, how can we do this?

i came across about the asteroids and their chances of drastic effects to our planet in nearby (or far away) future.Additionally i read about the techniques which handled by several space organizations including NASA to combat the effects.These things are well known to you....but i have small doubt regarding this.,.since we have cloud of satellites over globe,why we not able study these celestial objects in an efficient manner?even the techniques which handled by NASA didn't completed the desired it possible to destruct the asteroids in space itself,if it so what are the drawbacks?