Pattern formation is a natural property of nonlinear and non-equilibrium dynamical systems. Geophysical examples of such systems span practically all observable length scales, from rhythmic banding of chemical species within a single mineral crystal, to the morphology of cusps and spits along hundreds of kilometres of coastlines. This article briefly introduces the general principles of pattern formation and argues how they can be applied to open problems in the Earth sciences. Particular examples are then discussed, which summarize the contents of the rest of this Theme Issue.