Welcome to M635 Pattern Formation. From the stripes of a zebra and the spots on a leopard’s back to the ripples on a sandy beach or desert dune, regular patterns arise everywhere in nature. The appearance and evolution of these phenomena has been a focus of recent research activity across several disciplines that include: Physics, Biology, Chemistry, and Engineering. This course will provide an introduction to the range of mathematical theory and methods used to analyze and explain these often intrincate and beautiful patterns. Using several different approaches, from group theoretic methods to amplitude equations, the course will also provide insight behind the selection mechanisms of one pattern over another. Examples from Physics, Chemistry and Biology will motivate the mathematical models and theory.