Abstract. Language is striking in its systematic structure at all levels of description. By exhibiting combinatoriality and compositionality, each utterance in a language does not stand alone, but rather exhibits a network of dependencies on the other utterances in that language. Where does this structure come from? Why is language systematic, and where else might we expect to find this kind of systematicity in nature? In this talk, I will propose a simple hypothesis that systematic structure is the inevitable result of a suite of behaviours being transmitted by iterated learning. Iterated learning is a mechanism of cultural evolution in which behaviours persist by being learned through observation of that behaviour in another individual who acquired it in the same way. I will survey a wide range of lab studies of iterated learning, in which the cultural evolution of sets of behaviours is experimentally recreated. These studies include everything from artificial language learning tasks and sign language experiments, to more abstract behaviours like sequence learning, and have recently even been extended to other species. I will conclude by suggesting that these cultural evolution experiments provide clear predictions about where we should expect to see structure in behaviour, and what form that structure might take.