Many cells possess epithelial–mesenchymal plasticity (EMP), which allows them to shift reversibly between adherent, static and more detached, migratory states. These changes in cell behaviour are driven by the programmes of epithelial–mesenchymal transition (EMT) and mesenchymal–epithelial transition (MET), both of which play vital roles during normal development and tissue homeostasis. However, the aberrant activation of these processes can also drive distinct stages of cancer progression, including tumour invasiveness, cell dissemination and metastatic colonization and outgrowth. This review examines emerging common themes underlying EMP during tissue morphogenesis and malignant progression, such as the context dependence of EMT transcription factors, a central role for partial EMTs and the nonlinear relationship between EMT and MET. This article is part of a discussion meeting issue ‘Contemporary morphogenesis'.