These lectures are based on joint work with Anton Alekseev. Motivated by recent developments in physics (Jackiw-Teitelboim gravity), we consider infinite-dimensional moduli spaces of conformally compact hyperbolic metrics on surfaces with boundary, up to diffeomorphisms fixing the boundary. We show that these spaces carry natural symplectic structures, and are examples of Hamiltonian Virasoro spaces for the action of the diffeomorphisms of the boundary. *Lecture 1* will be an introduction to the Virasoro Lie algebra and its coadjoint orbits. *Lecture 2* will cover some basics on conformally compact metrics and their relation with flat connections. *Lecture 3* will explain our construction of the symplectic form and computation of the Virasoro moment map. Using Fenchel-Nielsen parameters we obtain a version of Wolpert's formula for the symplectic form, leading to global Darboux coordinates.