  We continue our study of heterotic compactifications on non-Kahler complex
manifolds with torsion. We give further evidence of the consistency of the
six-dimensional manifold presented earlier and discuss the anomaly cancellation
and possible supergravity description for a generic non-Kahler complex manifold
using the newly proposed superpotential. The manifolds studied in our earlier
papers had zero Euler characteristics. We construct new examples of non-Kahler
complex manifolds with torsion in lower dimensions, that have non-zero Euler
characteristics. Some of these examples are constructed from consistent
backgrounds in F-theory and therefore are solutions to the string equations of
motion. We discuss consistency conditions for compactifications of the
heterotic string on smooth non-Kahler manifolds and illustrate how some results
well known for Calabi-Yau compactifications, including counting the number of
generations, apply to the non-Kahler case. We briefly address various issues
regarding possible phenomenological applications.
