In recent years, the phenomenon of cumulative cultural evolution (CCE) has become the focus of major research interest in biology, psychology and anthropology. Some researchers argue that CCE is unique to humans and underlies our extraordinary evolutionary success as a species. Others claim to have found CCE in non-human species. Yet others remain sceptical that CCE is even important for explaining human behavioural diversity and complexity. These debates are hampered by multiple and often ambiguous definitions of CCE. Here, we review how researchers define, use and test CCE. We identify a core set of criteria for CCE which are both necessary and sufficient, and may be found in non-human species. We also identify a set of extended criteria that are observed in human CCE but not, to date, in other species. Different socio-cognitive mechanisms may underlie these different criteria. We reinterpret previous theoretical models and observational and experimental studies of both human and non-human species in light of these more fine-grained criteria. Finally, we discuss key issues surrounding information, fitness and cognition. We recommend that researchers are more explicit about what components of CCE they are testing and claiming to demonstrate.