Much of perception, learning and high-level cognition involves finding patterns in data. But there are always infinitely many patterns compatible with any finite amount of data. How does the cognitive system choose 'sensible' patterns? A long tradition in epistemology, philosophy of science, and mathematical and computational theories of learning argues that patterns 'should' be chosen according to how simply they explain the data. This article reviews research exploring the idea that simplicity drives a wide range of cognitive processes. We outline mathematical theory, computational results and empirical data that underpin this viewpoint.