In this talk I will describe a class of topological dynamical systems called Smale spaces. A Smale space (X, \phi) consists of a compact metric space X together with an expansive homeomorphism \phi: X \to X that satisfies certain local properties. To each such system one may associate C*-algebras which attempt to capture the interesting structure of the system. I will also discuss my recent work with Robin Deeley, which shows how these C*-algebras fit into the classification programme for simple separable nuclear C*-algebras.