Abstract. In topological dynamics, one considers the continuous actions of a topological group on a compact space. We will be most interested in minimal actions, those for which every orbit is dense, and Polish groups, groups whose underlying topology is separable and completely metrizable. For countable discrete groups, we show in a precise sense that minimal actions can be wildly complicated. By contrast, for automorphism groups of countable structures, the work of Kechris, Pestov, and Todorcevic establishes a connection between groups whose minimal actions are simple and theorems in finite Ramsey theory. We show that this is in fact the only reason an automorphism group can have well-behaved minimal actions. With this precise correspondence in mind, we then consider possible dynamical formulations of infinite Ramsey theorems.