We will discuss the classical Euclidean, spherical and hyperbolic spaces and study manifolds that have geometric structures locally modeled on them. Examples will be given.

Mostow’s Rigidity Theorem says that a closed 3-manifold M admits at most one hyperbolic structure up to isometry. If M does admit such a structure, can one effectively describe its geometry (e.g., its volume, diameter or more general `shape’) in terms of the topology of M? We will survey a number of results in this theme.