Syntax (rules for combining words or elements) and semantics (meaning of expressions) are two pivotal features of human language, and interaction between them allows us to generate a limitless number of meaningful expressions. While both features were traditionally thought to be unique to human language, research over the past four decades has revealed intriguing parallels in animal communication systems. Many birds and mammals produce specific calls with distinct meanings, and some species combine multiple meaningful calls into syntactically ordered sequences. However, it remains largely unclear whether, like phrases or sentences in human language, the meaning of these call sequences depends on both the meanings of the component calls and their syntactic order. Here, leveraging recently demonstrated examples of meaningful call combinations, we introduce a framework for exploring the interaction between syntax and semantics (i.e. the syntax-semantic interface) in animal vocal sequences. We outline methods to test the cognitive mechanisms underlying the production and perception of animal vocal sequences and suggest potential evolutionary scenarios for syntactic communication. We hope that this review will stimulate phenomenological studies on animal vocal sequences as well as experimental studies on the cognitive processes, which promise to provide further insights into the evolution of language. This article is part of the theme issue 'What can animal communication teach us about human language?'