Superstrings with intrinsic torsion

We analyze the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form R^1,9-d×Md, in the common Neveu-Schwarz--Neveu-Schwarz (NS-NS) sector of type II string theory and also type I or heterotic string theory. The results are phrased in terms of the intrinsic torsion of G structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalized calibrations naturally appear since the geometries can always arise as solutions describing NS or type I or heterotic fivebranes wrapping calibrated cycles. Some new solutions are presented. In particular we find $d=6$ examples with a fibered structure which preserve N=1,2,3 supersymmetry in type II and include compact type I or heterotic geometries.