  Recently Witten proposed to consider elliptic genus in $N=2$ superconformal
field theory to understand the relation between $N=2$ minimal models and
Landau-Ginzburg theories. In this paper we first discuss the basic properties
satisfied by elliptic genera in $N=2$ theories. These properties are confirmed
by some fundamental class of examples. Then we introduce a generic procedure to
compute the elliptic genera of a particular class of orbifold theories, {\it
i.e.\/} the ones orbifoldized by $e^{2\pi iJ_0}$ in the Neveu-Schwarz sector.
This enables us to calculate the elliptic genera for Landau-Ginzburg orbifolds.
When the Landau-Ginzburg orbifolds allow an interpretation as target manifolds
with $SU(N)$ holonomy we can compare the expressions with the ones obtained by
orbifoldizing tensor products of $N=2$ minimal models. We also give sigma model
expressions of the elliptic genera for manifolds of $SU(N)$ holonomy.
