With R. Heluani and M. Szczesny. **Supersymmetry of the
Chiral deRham Complex**.

We present a superfield formulation of the chiral de Rham complex
(CDR) of Malikov-Schechtman-Vaintrob in the setting of a general
smooth manifold, and use it to endow CDR with superconformal
structures of geometric origin. Given a Riemannian metric, we
construct an N=1 structure on CDR (action of the N=1
super--Virasoro, or Neveu--Schwarz, algebra). If the metric is
K"ahler, and the manifold Ricci-flat, this is augmented to an N=2
structure. Finally, if the manifold is hyperk"ahler, we obtain an
N=4 structure. The superconformal structures are constructed
directly from the Levi-Civita connection. These structures provide
an analog for CDR of the extended supersymmetries of nonlinear
sigma-models.