Verbitsky's Global Torelli theorem has been one of the most important advances in the theory of holomorphic symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a version of the Global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly rich. Our results are obtained through the interplay of Hodge theory, deformation theory, and a further example of Verbitsky's technique which might go under the name "how to deduce beautiful consequences from ugly behavior of moduli spaces".