  We show that non-trivial SU(3) structures can be constructed on large classes
of Calabi-Yau three-folds. Specifically, we focus on Calabi-Yau three-folds
constructed as complete intersections in products of projective spaces,
although we expect similar methods to apply to other constructions and also to
Calabi-Yau four-folds. Among the wide range of possible SU(3) structures we
find Strominger-Hull systems, suitable for heterotic or type II string
compactifications, on all complete intersection Calabi-Yau manifolds. These
SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed
three-form flux which needs to be supported by source terms in the associated
Bianchi identity. We discuss the possibility of finding such source terms and
present first steps towards their explicit construction. Provided suitable
sources exist, our methods lead to Calabi-Yau compactifications of string
theory with a non Ricci-flat, physical metric which can be written down
explicitly and in analytic form.
