  We study the behavior of Dirac cohomology under Howe's
$\Theta$-correspondence in the case of complex reductive dual pairs. More
precisely, if $(G_1,G_2)$ is a complex reductive dual pair with $G_1$ and $G_2$
viewed as real groups, we describe those Harish-Chandra modules $\pi_1$ of
$G_1$ with nonzero Dirac cohomology whose $\Theta$-liftings $\Theta(\pi_1)$
still have nonzero Dirac cohomology. In this case, we compute explicitly the
Dirac cohomology of $\Theta(\pi_1)$.
