Human communities teem with a variety of social norms. In order to change unjust and harmful social norms, it is crucial to identify the psychological processes that give rise to them. Most researchers take it for granted that social norms are uniquely human. By contrast, we approach this matter from a comparative perspective, leveraging recent research on animal social behaviour. While there is currently only suggestive evidence for norms in nonhuman communities, we argue that human social norms are likely produced by a wide range of mechanisms, many of which we share with nonhuman animals. Approaching this variability from a comparative perspective can help norm researchers expand and reframe the range of hypotheses they test when attempting to understand the causes of socially normative behaviours in humans. First, we diagnose some of the theoretical obstacles to developing a comparative science of social norms, and offer a few basic constructs and distinctions to help norm researchers overcome these obstacles. Then we develop a six-dimensional model of the psychological and social factors that contribute to variability in both human and potential nonhuman norms. This article is part of the theme issue 'Social norm change: drivers and consequences'.