In 1872 F. Klein, based on ideas of S. Lie,proposed that the classical geometries be understood via the theory of transformation groups and homogeneous spaces. Later in the 1930's, influenced by ideas of E. Cartan, C. Ehresmann and J.H.C. Whitehead formulated the question of which topologies could support local coordinate systems modeled on a Klein geometry. In the 1970's Thurston formulated his 3-dimensional geometrization conjecture (now a theorem due to Perelman) in terms of locally homogeneous Riemannian structures.
For example, the topology of the 2-sphere is incompatible with ordinary plane Euclidean geometry -- no metrically accurate world atlas exists. In contrast, the 2-torus admits many Euclidean structures, and their isometry classes form a space with a rich (hyperbolic) geometry of its own. In this talk, I will survey this question in a general context and describe some notable successful classification problems in dimension 2 and 3. We develop the theme that the moduli space of such structures admits a rich geometry of its own, and supports highly nontrivial and natural Hamiltonian dynamical systems.