Syntax is habitually named as what sets human language apart from other communication systems, but how did it evolve? Comparative research on animal behaviour has contributed in important ways, with mainly three sets of data. First, animals have been subjected to artificial grammar tasks, based on the hypothesis that human syntax has evolved through advanced computational capacity. In these experiments humans generally outperform animals, but there are questions about validity, as experimental stimuli are (deliberately) kept devoid of semantic content. Second, animal communication has been compared in terms of the surface structures with the aim of developing a typology of animal syntax, based on the hypothesis that syntax is an evolutionary solution to the constraints of small signal repertoires. A wide range of combinatorial phenomena has been described, mainly in nonhuman primates, but there is little support for the hypothesis that syntax has emerged due to repertoire size constraints. A third way of studying the evolution of syntax is to compare how animals perceive and communicate about external events, the mental deep structure of syntax. Human syntax is closely aligned with how we perceive events in terms of agency, action and patience, each with subsidiary functions. The event perception hypothesis has been least explored in animals and requires a serious research programme.