Human expression is diverse and multi-faceted, ranging from ordinary language use to painting, from exaggerated displays of affection to micro-movements that aid coordination. Here we present and defend the claim that this expressive diversity is united by an interrelated suite of cognitive capacities, the evolved function of which is the expression and recognition of informative intentions. We describe how evolutionary dynamics normally leash communication to narrow domains of statistical mutual benefit, and how they are unleashed in humans. The relevant cognitive capacities are cognitive adaptations to living in a partner choice social ecology; and they are, correspondingly, part of the ordinarily developing human cognitive phenotype, emerging early and reliably in ontogeny. In other words, we identify distinctive features of our species’ social ecology that can explain how and why humans evolved the cognitive capacities that, in turn, lead to massive diversity in means and modes of expression. We make relevant cross-species comparisons, describe how the relevant cognitive capacities can evolve in a gradual manner, and survey how unleashed expression facilitates not only the evolution of languages, but novel behaviour in other domains too, focusing on the examples of joint action, punishment, and the arts. We aim to help reorient cognitive pragmatics, as a phenomenon that is not a supplement to linguistic communication and on the periphery of language science, but rather the foundation of many of the most of the most distinctive features of human behaviour and societies.