The rotation rates of asteroids, which are deduced from periodic fluctuations in their brightnesses, are controlled by mutual collisions. The link between asteroid spin and collision history is usually made with reference to impact experiments on centimetre-scale targets, where material strength governs the impact response. Recent work, however, indicates that for objects of the size of most observed asteroids (> or = 1 km in diameter), gravity rather than intrinsic strength controls the dynamic response to collisions. Here we explore this idea by modelling the effect of impacts on large gravitating bodies. We find that the fraction of a projectile's angular momentum that is retained by a target asteroid is both lower and more variable than expected from laboratory experiments, with spin evolution being dominated by 'catastrophic' collisions that eject approximately 50 per cent of the target's mass. The remnant of an initially non-rotating silicate asteroid that suffers such a collision rotates at a rate of approximately 2.9 per day, which is close to the observed mean asteroid rotation rate of approximately 2.5 d-1. Moreover, our calculations suggest that the observed trend in the mean spin frequency for different classes of asteroids (2.2 d-1 for C-type asteroids, 2.5 d-1 for S-type, and 4.0 d-1 for M-type) is due to increasing mean density, rather than increasing material strength.