Tiling spaces, and the dynamics induced by translation, have connections to many areas of mathematics. One dimensional tiling spaces generalize symbolic dynamics. Substitution tiling spaces (in one or higher dimensions) model expanding attractors. The dynamical properties of higher dimensional tiling spaces describe the diffraction properties of physical quasicrystals. In this talk, I'll review the dynamics and topology of tiling spaces and then present some new results on understanding homeomorphisms between tilings spaces. To wit: under some mild assumptions, every homeomorphism of tiling spaces is the composition of three maps: a self-map homotopic to the identity, a "shape change" that preserves the combinatorics of the tilings but distorts the shapes and sizes of the tiles, and a local relabeling.