  We formulate general conjectures about the relationship between the A-model
connection on the cohomology of a $d$-dimensional Calabi-Yau complete
intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety
${\bf P}_{\Sigma}$ and the system of differential operators annihilating the
special hypergeometric function $\Phi_0$ depending on the fan $\Sigma$. In this
context, the Mirror Symmetry phenomenon can be interpreted as the following
twofold characterization of the series $\Phi_0$. First, $\Phi_0$ is defined by
intersection numbers of rational curves in ${\bf P}_{\Sigma}$ with the
hypersurfaces $V_i$ and their toric degenerations. Second, $\Phi_0$ is the
power expansion near a boundary point of the moduli space of the monodromy
invariant period of the holomorphic differential $d$-form on an another
Calabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalized
hypergeometric series, we propose a general construction for Mirrors $V'$ of
$V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.
