I will discuss the quantization of one of the simplest theories of gravity, three-dimensional Einstein gravity with negative cosmological constant. In particular I will describe a precise reformulation of AdS_3 quantum gravity in terms of a novel topological quantum field theory based on the quantization of the Teichmuller space of Riemann surfaces that we refer to as "Virasoro TQFT." This proposal clarifies the relationship between pure AdS_3 quantum gravity and (two copies of) SL(2,R) Chern-Simons theory, and resolves some well-known issues with this lore. I will argue that moreover this proposal provides a practical framework for the computation of 3d gravity partition functions, and will describe how it elucidates various aspects of the holographic dual.