  We discuss the mathematical properties of six--dimensional non--K\"ahler
manifolds which occur in the context of ${\cal N}=1$ supersymmetric heterotic
and type IIA string compactifications with non--vanishing background H--field.
The intrinsic torsion of the associated SU(3) structures falls into five
different classes. For heterotic compactifications we present an explicit
dictionary between the supersymmetry conditions and these five torsion classes.
We show that the non--Ricci flat Iwasawa manifold solves the supersymmetry
conditions with non--zero H--field, so that it is a consistent heterotic
supersymmetric groundstate.
