  We study the class of compact complex manifolds whose first Chern class
vanishes in the Bott-Chern cohomology. This class includes all manifolds with
torsion canonical bundle, but it is strictly larger. After making some
elementary remarks, we show that a manifold in Fujiki's class C with vanishing
first Bott-Chern class has torsion canonical bundle. We also give some examples
of non-Kahler Calabi-Yau manifolds, and discuss the problem of defining and
constructing canonical metrics on them.
