A Smale space, as defined by David Ruelle, is a topological dynamical system having a type of hyperbolic structure. Examples include Anosov diffeomeorphisms, certain solenoids, certain substitution tiling systems, shifts of finite type and basic sets from Smale’s axiom A system. It is possible to construct C ∗ -algebras from such systems. In the case of shifts of finite type, these are AFalgebras. The talk will describe the basic ideas of Smale spaces, along with examples, the construction of the C*-algebras and various properties which they possess. In particular, there is a K-theoretic duality between a pair of them. Finally, the talk will describe a homology theory for Smale spaces which is closely related to the K-theory of the C ∗ -algebras