  We systematically analyse the necessary and sufficient conditions for the
preservation of supersymmetry for bosonic geometries of the form R^{1,9-d}
\times M_d, in the common NS-NS sector of type II string theory and also type
I/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. Generalised calibrations
naturally appear since the geometries always admit NS or type I/heterotic
fivebranes wrapping calibrated cycles. Some new solutions are presented. In
particular we find d=6 examples with a fibred structure which preserve N=1,2,3
supersymmetry in type II and include compact type I/heterotic geometries.
