  The phenomenon of black hole thermodynamics raises several deep issues which
any proper theory of quantum gravity must confront: to what extent does the
inclusion of the back-reaction alter the thermal character of the radiation,
how can the entropy be understood from a microscopic standpoint, what is the
ultimate fate of an evaporating black hole, and is the outcome reconcilable
with unitary time evolution in quantum mechanics?
  In the first part of this thesis, we address the issue of determining what
the actual emission spectrum from a black hole is, once the gravitational field
of the emitted quanta is included in a quantum mechanical manner. To make the
problem tractable, we employ two important approximations: we quantize only the
s-wave sector of the full theory, and we consider only single particle
emission. By proceeding in the framework of a Hamiltonian path integral
description of this system, we are able to integrate out the gravitational
field, thereby obtaining an effective action depending only on the matter
degrees of freedom. This effective action can then be second quantized in terms
of new, corrected, mode solutions thus enabling the calculation of the emission
spectrum from modified Bogoliubov coefficients. The results are particularly
interesting in the case of emission from Reissner-Nordstrom black holes, since
in the extremal limit our results are dramatically different from what a naive,
and incorrect, semi-classical calculation would yield.
  The other major topic which we discuss is the dynamics of quantum fields on
background geometries which undergo quantum tunneling. An example of such a
system which has important implications for both cosmology and quantum gravity
in general, is the tunneling of a false vacuum bubble leading to the creation
