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.

58 pages, LaTeX; v2: New section on solutions including an example with N=3 supersymmetry and discussion of heterotic compactifications. Details on conventions and references added. v3: added an explicit example of non-integrable product structure in Appendix C; some typos fixed