In this paper the author presents evidences showing that for most of the asteroids the motions are stable in the sense that they never approach major planets very closely and explains about mechanisms to avoid very close approaches by investigating the variations due to the secular perturbations of the eccentricities as functions of the arguments of perihelion, particularly, for asteroids with high eccentricities and inclinations. It is believed that some kinds of dynamical evolution processes have made the asteroid motions stable. The author shows also that there were some kinds of collisions among asteroids in the past which produced families and present distribution of asteroids as there are very faint asteroids only near Kirkwood gaps.