  Recent discoveries in differential topology are reviewed in light of their
possible implications for spacetime models and related subjects in theoretical
physics. Although not often noted, a particular smoothness (differentiability)
structure must be imposed on a topological manifold before geometric or other
structures of physical interest can be discussed. The recent discoveries of
interest here are of various surprising ``exotic'' smoothness structures on
topologically trivial manifolds such as ${S^7}$ and ${\bf R^4}$. Since no two
of these are diffeomorphic to each other, each such manifold represents a
physically distinct model of topologically trivial spacetime. That is, these
are not merely different coordinate representations of a given spacetime. The
path to such structures intertwines many branches of mathematics and
theoretical physics (Yang-Mills and other gauge theories). An overview of these
topics is provided, followed by certain results concerning the geometry and
physics of such manifolds. Although exotic ${\bf R^4}$'s cannot be effectively
exhibited by finite constructions, certain existence and non-existence results
can be stated. For example, it is shown that the ``exoticness'' can be confined
to a time-like world tube, providing a possible model for an exotic source.
Other suggestions and conjectures for future research are made.
