  Quantum gravity (QG) on two-dimensional anti-de Sitter spacetime (AdS_2)
takes always the form of a chiral conformal field theory (CFT). However, the
actual content of the CFT, and in particular its central charge, depends on the
background values of the dilaton and Maxwell field. We review the main features
of AdS_2 QG with linear dilaton and of AdS_2 QG with constant dilaton and
Maxwell field. We also show that the 3D charged Banados-Teitelboim-Zanelli
black hole interpolates between these two versions of AdS_2 QG. Applications to
the computation of the microscopic entropy of black holes are also discussed.
