Explore hyperbolic surfaces and their Teichmüller spaces in this advanced lecture from the Geometry, Groups and Dynamics program at the International Centre for Theoretical Sciences. Delve into key concepts like the collar lemma, surface group representations, and proper discontinuous actions. Examine the relationship between hyperbolic metrics and geometric structures modeled on PSL2(R) and H2. Investigate developing maps, holonomy representations, and their properties. Gain insights into the continuous nature of these mathematical structures through detailed proofs and examples presented over the course of 71 minutes.