  We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped
with a metric connection $\nabla$ with skew torsion $T\in\Lambda^3M$ by means
of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac
operator with torsion is found that generalizes Friedrich's classical
Riemannian estimate. We also determine a novel twistor and Killing equation
with torsion and use it to discuss the case in which the minimum is attained in
the bound.
