Most language usage is interactive, involving rapid turn-taking. The turn-taking system has a number of striking properties: turns are short and responses are remarkably rapid, but turns are of varying length and often of very complex construction such that the underlying cognitive processing is highly compressed. Although neglected in cognitive science, the system has deep implications for language processing and acquisition that are only now becoming clear. Appearing earlier in ontogeny than linguistic competence, it is also found across all the major primate clades. This suggests a possible phylogenetic continuity, which may provide key insights into language evolution.