  The aim of this note is to define certain sheaves of vertex algebras on
smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$,
we construct a sheaf $\Omega^{ch}_X$, called the {\bf chiral de Rham complex}
of $X$. It is a sheaf of vertex algebras in the Zarisky (or classical)
topology, It comes equipped with a $\BZ$-grading by {\it fermionic charge}, and
the {\it chiral de Rham differential} $d_{DR}^{ch}$, which is an endomorphism
of degree 1 such that $(d_{DR}^{ch})^2=0$. One has a canonical embedding of the
usual de Rham complex $(\Omega_X, d_{DR})\hra (\Omega_X^{ch}, d_{DR}^{ch})$
which is a quasiisomorphism.
  If $X$ is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some
$X$ (for example, for curves or for the flag spaces $G/B$), one can construct
also a purely even analogue of this sheaf, a {\it chiral structure sheaf}
$\CO^{ch}_X$. For the projective line, the space of global sections of the last
sheaf is the irreducible vacuum $\hsl(2)$-module on the critical level.
