We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection∇ with torsion H, the NS⊗NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups K R 8 in Spin(9, 1), for K = Spin(7), SU (4), Sp(2), SU (2) × SU (2) and {1}, and the Killing spinors of the timelike backgrounds have stability subgroups G 2 , SU (3), SU (2) and {1}. The former admit a single null∇-parallel vector field while the latter admit a timelike and two, three, five and nine spacelike∇-parallel vector fields, respectively. The spacetime of the null backgrounds is a Lorentzian two-parameter family of Riemannian manifolds B with skew-symmetric torsion. If the rotation of the null vector field vanishes, the holonomy of the connection with torsion of B is contained in K. The spacetime of time-like backgrounds is a principal bundle P with fibre a Lorentzian Lie group and base space a suitable Riemannian manifold with skew-symmetric torsion. The principal bundle is equipped with a connection λ which determines the non-horizontal part of the spacetime metric and of H. The curvature of λ takes values in an appropriate Lie algebra constructed from that of K. In addition dH has only horizontal components and contains the Pontrjagin class of P . We have computed in all cases the Killing spinor bilinears, expressed the fluxes in terms of the geometry and determine the field equations that are implied by the Killing spinor equations.