This theme issue explores how and why behavioural innovation occurs, and the consequences of innovation for individuals, groups and populations. A vast literature on human innovation exists, from the development of problem-solving in children, to the evolution of technology, to the cultural systems supporting innovation. A more recent development is a growing literature on animal innovation, which has demonstrated links between innovation and personality traits, cognitive traits, neural measures, changing conditions, and the current state of the social and physical environment. Here, we introduce these fields, define key terms and discuss the potential for fruitful exchange between the diverse fields researching innovation. Comparisons of innovation between human and non-human animals provide opportunities, but also pitfalls. We also summarize some key findings specifying the circumstances in which innovation occurs, discussing factors such as the intrinsic nature of innovative individuals and the environmental and socio-ecological conditions that promote innovation, such as necessity, opportunity and free resources. We also highlight key controversies, including the relationship between innovation and intelligence, and the notion of innovativeness as an individual-level trait. Finally, we discuss current research methods and suggest some novel approaches that could fruitfully be deployed.