  We show that the supersymmetry transformations for type II string theories on
six-manifolds can be written as differential conditions on a pair of pure
spinors, the exponentiated Kahler form exp(iJ) and the holomorphic form Omega.
The equations are explicitly symmetric under exchange of the two pure spinors
and a choice of even or odd-rank RR field. This is mirror symmetry for
manifolds with torsion. Moreover, RR fluxes affect only one of the two
equations: exp(iJ) is closed under the action of the twisted exterior
derivative in IIA theory, and similarly Omega is closed in IIB. Modulo a
different action of the B-field, this means that supersymmetric SU(3)-structure
manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An
equivalent, and somewhat more conventional, description is given as a set of
relations between the components of intrinsic torsions modified by the NS flux
and the Clifford products of RR fluxes with pure spinors, allowing for a
classification of type II supersymmetric vacua on six-manifolds. We find in
particular that supersymmetric six-manifolds are always complex for IIB
backgrounds while they are twisted symplectic for IIA.
