Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler
manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called {\em real bisectional curvature}
for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$,
and when the metric is non-K\"ahler, it is slightly stronger than $H$. We classify compact Hermitian manifolds
with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case.
The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric
is only Hermitian but has nonpositive real bisectional curvature.