  We prove that any compact complex homogeneous space with vanishing first
Chern class after an appropriate deformation of the complex structure admits a
homogeneous Calabi-Yau with torsion structure, provided that it also has an
invariant volume form. A description of such spaces among the homogeneous
C-spaces is given as well as many examples and a classification in the
3-dimensional case. We calculate the cohomology ring of some of the examples
and show that in dimension 14 there are infinitely many simply-connected spaces
with the same Hodge numbers and torsional Chern classes admitting such
structure. We provide also an example solving the Strominger's equations in
heterotic string theory.
