Smale spaces were defined by David Ruelle in the 1970's as topological models for the typically fractal-like hyperbolic nonwandering sets of Stephen Smale's Axiom A systems. A Smale space is a compact metric space together with a homeomorphism implementing hyperbolic dynamics and a local product structure. Prototype examples are the topological Markov chains, aperiodic substitution tilings and hyperbolic toral automorphisms. This talk will give an example-driven introduction to Smale spaces with a focus on their dimension theory, which can be studied using Markov partitions and Ahlfors regular measures. Finally, I will briefly mention how the dimension theory of a Smale space is related to fine analytic properties of the operator algebras encoding the stable and unstable foliations on it.