Teichmuller spaces for geometric structures and the mapping class group action

The Teichmuller space of geometric structures of a given type is a quotient of the (generally, infinite-dimensional) space of geometric structures by the group of isotopies, that is, by the connected component of the diffeomorphism group. In several important qand smooth.uestions, such as for symplectic, hyperkahler, Calabi-Yau, G2 structures, this quotient is finite-dimenisional and even smooth. The mapping class group acts on the Teichmuller space by natural diffeomorphisms, and this action is in many important situations ergodic (in particular, it has dense orbits), bringing strong consequences for the geometry. I would describe the Teichmuller space for the best understood cases, such as symplectic and hyperkahler manifolds, and give a few geometric applications.