A Smale space $(X, \phi)$ is a hyperbolic topological dynamical system with the property that $X$ can be locally decomposed into expanding and contracting coordinates. Examples include shifts of finite type, hyperbolic toral automorphisms and certain aperiodic substitution tilings.

Given an irreducible Smale space, one may associate $C*4-algebras, and it is of interest to $C*$-algebraists what kind of structures might arise. Conversely, this association allows one to use operator algebraic techniques to study the underlying Smale spaces.

In this talk I will introduce Smale spaces and give a number of examples. I will give a brief overview of the associated $C*$-algebras and discuss the interactions between these $C*$-algebras and their underlying Smale space. Time permitting, I will also discuss my own ongoing research, which is joint work with Robin Deeley.