  We study the open string extension of the mirror map for N=1 supersymmetric
type II vacua with D-branes on non-compact Calabi-Yau manifolds. Its definition
is given in terms of a system of differential equations that annihilate certain
period and chain integrals. The solutions describe the flat coordinates on the
N=1 parameter space, and the exact disc instanton corrected superpotential on
the D-brane world-volume. A gauged linear sigma model for the combined
open-closed string system is also given. It allows to use methods of toric
geometry to describe D-brane phase transitions and the N=1 K\"ahler cone.
Applications to a variety of D-brane geometries are described in some detail.
