A Poisson vertex algebra is a unital commutative associative algebra with a derivation and a $\lambda$-bracket; it has to satisfy the axioms of a Lie conformal algebra; and the $\lambda$-bracket and the multiplication form a Poisson algebra (i.e. satisfy the Leibniz rule). It is a Poisson analogue of a conformal Lie algebra.

A. De Sole, V. G. Kac, M.; Wakimoto, On classification of Poisson vertex algebras, Transform. Groups 15 (2010), no. 4, 883–907 MR2012a:17051doi

B. Bakalov, A. De Sole, Non-linear Lie conformal algebras with three generators, Selecta Math. (N.S.) 14 (2009), no. 2, 163–198 dpoMR2009m:17024

Alberto De Sole, Victor G. Kac, Daniele Valeri, Classical W-algebras and Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, arxiv/1207.6286

Gaywalee Yamskulna, Vertex Poisson algebras associated with Courant algebroids and their deformations; I, math.QA/0509122

A. De Sole, Poisson vertex algebras in the theory of Hamiltonian equations, Talk at Algebraic Lie Theory, Newton Inst. U. Cambridge 2009, video

On non-local Poisson vertex algebras with applications to the study of integrability of Hamiltonian PDEs in