  The recent result of Strominger, Yau and Zaslow relating mirror symmetry to
the quantum field theory notion of T-duality is reinterpreted as providing a
way of geometrically characterizing which Calabi-Yau manifolds have mirror
partners. The geometric description---that one Calabi-Yau manifold should serve
as a compactified, complexified moduli space for special Lagrangian tori on the
other Calabi-Yau manifold---is rather surprising. We formulate some precise
mathematical conjectures concerning how these moduli spaces are to be
compactified and complexified, as well as a definition of geometric mirror
pairs (in arbitrary dimension) which is independent of those conjectures. We
investigate how this new geometric description ought to be related to the
mathematical statements which have previously been extracted from mirror
symmetry. In particular, we discuss how the moduli spaces of the `mirror'
Calabi-Yau manifolds should be related to one another, and how appropriate
subspaces of the homology groups of those manifolds could be related. We treat
the case of K3 surfaces in some detail.
