  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.
