  Orbifold elliptic genus and elliptic genus of singular varieties are
introduced and relation between them is studied. Elliptic genus of singular
varieties is given in terms of a resolution of singularities and extends the
elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties
introduced earlier. Orbifold elliptic genus is given in terms of the fixed
point sets of the action. We show that the generating function for this
orbifold elliptic genus $\sum Ell_{orb}(X^n,\Sigma_n)p^n$ for symmetric groups
$\Sigma_n$ acting on $n$-fold products coincides with the one proposed by
Dijkgraaf, Moore, Verlinde and Verlinde. Two notions of elliptic genera are
conjectured to coincide.
