  We consider non-Kaehler compact complex manifolds which are homogeneous under
the action of a compact Lie group of biholomorphisms and we investigate the
existence of special (invariant) Hermitian metrics on these spaces. We focus on
a particular class of such manifolds comprising the case of Calabi-Eckmann
manifolds and we prove the existence of an invariant Hermitian metric which is
Chern-Einstein, namely whose second Ricci tensor of the associated Chern
connection is a positive multiple of the metric itself. The uniqueness is also
discussed.
