2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148378This is a survey paper that also contains some new results. It will appear in the proceedings of the AMS summer research institute on Algebraic Geometry at Santa Cruz.We expanded section 7 and rewrote parts of section 10. We also did some editing and made some minor corrections. latex2e, 46 pagesAlgebraic Geometry14-02, 14H10, 32G15Mapping Class Groups and Moduli Spaces of Curvestext