Three hypotheses concerning the origin of asteroids were examined, vtz. that the asteroids are collisional fragments deriving from a few primaeval planetoids; that they are the remnants of a planet which disrupted; and that they are in process of forming a single body by random aggregation. Monte Carlo simulations and theoretical arguments were used to predict mass distributions and rotation periods. It was found that the accretion hypothesis was not in good agreement with observations; but that both the collisional fragmentation and exploded planet theories were consistent with the observed asteroidal rotations and mass distribution. Physical arguments were used to show that accretion within the asteroid belt, through the intermediary of jet stream formation, is probably not taking place, and that the exploded planet hypothesis meets with severe difficulties. It was concluded that, most probably, the asteroids are debris arising from the collisional shattering of a few primal bodies, some of which may still survive intact.