Dominance-based social hierarchies are common among teleost fishes. The rank of an animal greatly affects its behaviour, physiology and development. The outcome of fights for social dominance is affected by heritable factors and previous social experience. Divergent stress-coping styles have been demonstrated in a large number of teleosts, and fish displaying a proactive coping style have an advantage in fights for social dominance. Coping style has heritable components, but it appears to be largely determined by environmental factors, especially social experience. Agonistic behaviour is controlled by the brain's social decision-making network, and its monoaminergic systems play important roles in modifying the activity of this neuronal network. In this Review, we discuss the development of dominance hierarchies, how social rank is signalled through visual and chemical cues, and the neurobiological mechanisms controlling or correlating with agonistic behaviour. We also consider the effects of social interactions on the welfare of fish reared in captivity.