  If $X$ and $Y$ are a mirror pair of Calabi--Yau threefolds, mirror symmetry
should extend to an isomorphism between the type IIA string theory compactified
on $X$ and the type IIB string theory compactified on $Y$, with all
nonperturbative effects included. We study the implications which this proposal
has for the structure of the semiclassical moduli spaces of the compactified
type II theories. For the type IIB theory, the form taken by discrete shifts in
the Ramond-Ramond scalars exhibits an unexpected dependence on the $B$-field.
(Based on a talk at the Trieste Workshop on S-Duality and Mirror Symmetry.)
