  We use mirror symmetry to establish the first concrete arena of spacetime
topology change in string theory. In particular, we establish that the {\it
quantum theories} based on certain nonlinear sigma models with topologically
distinct target spaces can be smoothly connected even though classically a
physical singularity would be encountered. We accomplish this by rephrasing the
description of these nonlinear sigma models in terms of their mirror manifold
partners--a description in which the full quantum theory can be described
exactly using lowest order geometrical methods.
  We establish that, for the known class of mirror manifolds, the moduli space
of the corresponding conformal field theory requires not just two but {\it
numerous} topologically distinct Calabi-Yau manifolds for its geometric
interpretation. A {\it single} family of continuously connected conformal
theories thereby probes a host of topologically distinct geometrical spaces
giving rise to {\it multiple mirror manifolds}.
