The talk will focus on tilings of the plane that are almost periodic. Such tilings serve as models of ideal quasi-crystals and can have symmetry properties unseen in periodic tilings (which model ordinary crystals). I will recount the construction of the famous Penrose tiling and follow with an explanation of the concept of a tiling space. This is to frame some new results about linear and nonlinear symmetries of aperiodic tilings and about their topological rigidity. The main result is that, for self-similar tilings, the topology of the tiling space largely determines geometric, dynamical, and spectral properties of the tiling.