The notions of orbifold elliptic genus and elliptic genus of
singular varieties are introduced, and the relation between them is
studied. The 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. The orbifold elliptic genus is given in terms of the
fixed-point sets of the action. We show that the generating function
for the orbifold elliptic $\sum {\rm Ell\sp {orb}}(X\sp n,\Sigma\sb
n)p\sp n$ for symmetric groups $\Sigma\sb n$ acting on $n$-fold
products coincides with the one proposed by R. Dijkgraaf, G. Moore,
E. Verlinde, and H. Verlinde. The two notions of elliptic genera are
conjectured to coincide.