  Non-perturbative instanton corrections to the moduli space geometry of type
IIA string theory compactified on a Calabi-Yau space are derived and found to
contain order $e^{-1/g_s}$ contributions, where $g_s$ is the string coupling.
The computation reduces to a weighted sum of supersymmetric extremal maps of
strings, membranes and fivebranes into the Calabi-Yau space, all three of which
enter on equal footing. It is shown that a supersymmetric 3-cycle is one for
which the pullback of the K\"ahler form vanishes and the pullback of the
holomorphic three-form is a constant multiple of the volume element. Quantum
mirror symmetry relates the sum in the IIA theory over supersymmetric,
odd-dimensional cycles in the Calabi-Yau space to a sum in the IIB theory over
supersymmetric, even-dimensional cycles in the mirror.
