  The geometric construction of the functional integral over coset spaces
${\cal M}/{\cal G}$ is reviewed. The inner product on the cotangent space of
infinitesimal deformations of $\cal M$ defines an invariant distance and volume
form, or functional integration measure on the full configuration space. Then,
by a simple change of coordinates parameterizing the gauge fiber $\cal G$, the
functional measure on the coset space ${\cal M}/{\cal G}$ is deduced. This
change of integration variables leads to a Jacobian which is entirely
equivalent to the Faddeev-Popov determinant of the more traditional gauge fixed
approach in non-abelian gauge theory. If the general construction is applied to
the case where $\cal G$ is the group of coordinate reparametrizations of
spacetime, the continuum functional integral over geometries, {\it i.e.}
metrics modulo coordinate reparameterizations may be defined. The invariant
functional integration measure is used to derive the trace anomaly and
effective action for the conformal part of the metric in two and four
dimensional spacetime. In two dimensions this approach generates the
Polyakov-Liouville action of closed bosonic non-critical string theory. In four
dimensions the corresponding effective action leads to novel conclusions on the
importance of quantum effects in gravity in the far infrared, and in
particular, a dramatic modification of the classical Einstein theory at
cosmological distance scales, signaled first by the quantum instability of
classical de Sitter spacetime. Finite volume scaling relations for the
functional integral of quantum gravity in two and four dimensions are derived,
and comparison with the discretized dynamical triangulation approach to the
integration over geometries are discussed.
