Norms permeate human life. Most of our activities can be characterised by rules about what is appropriate, allowed, required, or forbidden – rules that are crucial in making us hyper-cooperative animals. This article examines the current cognitive-evolutionary account of ‘norm psychology’ and proposes an alternative that is better supported by evidence and better placed to promote interdisciplinary dialogue. The incumbent theory focusses on rules and claims that humans genetically inherit cognitive and motivational mechanisms specialised for processing these rules. The cultural evolutionary alternative defines normativity in relation to behaviour – compliance, enforcement, and commentary – and suggests that it depends on implicit and explicit processes. The implicit processes are genetically inherited and domain-general; rather than being specialised for normativity, they do many jobs in many species. The explicit processes are culturally inherited and domain-specific; they are constructed from mentalising and reasoning by social interaction in childhood. The cultural evolutionary, or ‘cognitive gadget’, perspective suggests that people alive today - parents, educators, elders, politicians, lawyers – have more responsibility for sustaining normativity than the nativist view implies. Our actions do not only shape and transmit the rules, they create in each new generation mental processes that can grasp the rules and put them into action.