Rich classes of geometric structures on manifolds are defined by coordinate atlases taking values in a fixed homogeneous space. The existence and classification of such structures leads to a moduli space, which itself is modelled on the algebraic variety of representations of the fundamental group in the automorphism group of the geometry. Topological symmetries lead to actions of mapping class groups on the moduli spaces, whose dynamics reflects the topology and the geometry. This talk will present various examples of the general classification problem, in dimensions 2 and 3.