i will discuss a new cohomology theory that extends $h$. cartan's cohomology theory of $g^*$ algebras. the latter is an algebraic abstraction of the topological equivariant cohomology theory for $g-$spaces, where $g$ is a compact lie group. cartan's theory, discovered in the 50s and further developed by others in the 90s, gave a de rham model for the topological equivariant cohomology, the same way ordinary de rham theory does for singular cohomology in a geometric setting. the chiral equivariant cohomology takes values in a vertex algebra and includes cartan's cohomology as a subalgebra. i will give a brief introduction to vertex algebras, and then discuss the construction of the new cohomology and some of the basic results and examples. this is a joint work with bong lian and bailin song.