  We construct a new equivariant cohomology theory for a certain class of
differential vertex algebras, which we call the chiral equivariant cohomology.
A principal example of a differential vertex algebra in this class is the
chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a
group action. The main idea in this paper is to synthesize the algebraic
approach to classical equivariant cohomology due to H. Cartan, with the theory
of differential vertex algebras, by using an appropriate notion of invariant
theory. We also construct the vertex algebra analogues of the Mathai-Quillen
isomorphism, the Weil and the Cartan models for equivariant cohomology, and the
Chern-Weil map. We give interesting cohomology classes in the new theory that
have no classical analogues.
