  We study two natural problems concerning the scalar and the Ricci curvatures
of the Bismut connection. Firstly, we study an analog of the Yamabe problem for
Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed
a conformal Hermitian structure on a compact complex manifold, there exists a
metric with constant Bismut scalar curvature in that class when the expected
constant scalar curvature is non-negative. A similar result is given in the
general case of Gauduchon connections. We then study an Einstein-type condition
for the Bismut Ricci curvature tensor on principal bundles over Hermitian
manifolds with complex tori as fibers. Thanks to this analysis we construct
explicit examples of Calabi--Yau with torsion Hermitian structures and prove a
uniqueness result for them.
