Language has been considered by many to be uniquely human. Numerous theories for how it evolved have been proposed but rarely tested. The articles in this theme issue consider the extent to which aspects of language, such as vocal learning, phonology, syntax, semantics, intentionality, cognition and neurobiological adaptations, are shared with other animals. By adopting a comparative approach, insights into the mechanisms and origins of human language can be gained. While points of agreement exist among the authors, conflicting viewpoints are expressed on several issues, such as the presence of proto-syntax in animal communication, the neural basis of the Merge operation, and the neurogenetic changes necessary for vocal learning. Future comparative research in animal communication has the potential to teach us even more about the evolution, neurobiology and cognitive basis of human language. This article is part of the theme issue 'What can animal communication teach us about human language?'