THE orbits of comets and near-Earth asteroids evolve chaotically, mainly in response to the gravitational influence of the planets. For comets, it is known that such perturbations can result in trajectories that either collide with or graze the Sun(1-3). Indeed, it has been calculated(3) that 6% of the known short-period comets (including comet Encke) will become Sun-grazers an a timescale of similar to 10(5) years. But there has been no previous indication that asteroids may suffer a similar fate. To address this question, we have integrated numerically the orbits of several near-Earth asteroids. We find that these asteroids can also undergo solar collisions, through several dynamical routes involving orbital resonances with the giant planets, on timescales of the order of 10(6) years. Of the 47 objects studied, we found 19 cases in which the asteroid collided with the Sun, implying that solar collisions are a more common fate than planetary collisions or ejection from the Solar System. Past Sun-grazing events may also have been recorded in the surface composition and spectral properties of some existing near-Earth asteroids.