Syntax has been found in animal communication but only humans appear to have generative, hierarchically structured syntax. How did syntax evolve? I discuss three theories of evolutionary transition from animal to human syntax: computational capacity, structural flexibility and event perception. The computation hypothesis is supported by artificial grammar experiments consistently showing that only humans can learn linear stimulus sequences with an underlying hierarchical structure, a possible by-product of computationally powerful large brains. The structural flexibility hypothesis is supported by evidence of meaning-bearing combinatorial and permutational signal sequences in animals, with sometimes compositional features, but no evidence for generativity or hierarchical structure. Again, animals may be constrained by computational limits in short-term memory but possibly also by limits in articulatory control and social cognition. The event categorization hypothesis, finally, posits that humans are cognitively predisposed to analyse natural events by assigning agency and assessing how agents impact on patients, a propensity that is reflected by the basic syntactic units in all languages. Whether animals perceive natural events in the same way is largely unknown, although event perception may provide the cognitive grounding for syntax evolution. This article is part of the theme issue ‘What can animal communication teach us about human language?’