  A midi-superspace model is a field theory obtained by symmetry reduction of a
parent gravitational theory. Such models have proven useful for exploring the
classical and quantum dynamics of the gravitational field. I present 3 recent
classes of results pertinent to canonical quantization of vacuum general
relativity in the context of midi-superspace models. (1) I give necessary and
sufficient conditions such that a given symmetry reduction can be performed at
the level of the Lagrangian or Hamiltonian. (2) I discuss the Hamiltonian
formulation of models based upon cylindrical and toroidal symmetry. In
particular, I explain how these models can be identified with parametrized
field theories of wave maps, thus a natural strategy for canonical quantization
is available. (3) The quantization of a parametrized field theory, such as the
midi-superspace models considered in (2), requires construction of a quantum
field theory on a fixed (flat) spacetime that allows for time evolution along
arbitrary foliations of spacetime. I discuss some recent results on the
possibility of finding such a quantum field theory.
