Cohomology of conformal algebras

The notion of a conformal algebra encodes an axiomatic description of the o perator product expansion of chiral fields in conformal field theory. On th e other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Li e algebras are those "based" on the punctured complex plane, such as the Vi rasoro algebra and loop Lie algebras. In the present paper we develop a coh omology theory of conformal algebras with coefficients in an arbitrary modu le. It possesses standards properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations fo r the most important examples.