  Given a compact complex algebraic variety with an effective action of a
finite group $G$, and a class $\alpha \in H^2(G,U(1))$, we introduce an
orbifold elliptic genus with discrete torsion $\alpha$, denoted
$Ell^{\alpha}_{orb}(X,G, q, y)$. We give an interpretation of this genus in
terms of the chiral de Rham complex attached to the orbifold $[X/G]$. If $X$ is
Calabi-Yau and $G$ preserves the volume form, $Ell^{\alpha}_{orb}(X,G, q, y)$
is a weak Jacobi form. We also obtain a formula for the generating function of
the elliptic genera of symmetric products with discrete torsion.
