  In this article, we study Hermitian manifolds whose Bismut connection has
parallel torsion, which will be called {\em Bismut torsion parallel manifolds,}
or {\em BTP} manifolds for brevity. We obtain a necessary and sufficient
condition characterizing BTP manifolds in terms of Bismut curvature tensor
alone (Theorem 1.1). We also present examples and discuss some general
properties for BTP manifolds, as well as give a classification result for
non-balanced BTP threefolds (Theorem 1.16).
