  Boundary conditions play a crucial role in the path-integral approach to
quantum gravity and quantum cosmology, as well as in the current attempts to
understand the one-loop semiclassical properties of quantum field theories.
Within this framework, one is led to consider boundary conditions completely
invariant under infinitesimal diffeomorphisms on metric perturbations. These
are part of a general scheme, which can be developed for Maxwell theory,
Yang-Mills Theory, Rarita-Schwinger fields and any other gauge theory. A
general condition for strong ellipticity of the resulting field theory on
manifolds with boundary is here proved, following recent work by the authors.
The relevance for Euclidean quantum gravity is eventually discussed.
