  We discuss the theory of Poisson vertex algebras and their generalizations in
relation to integrability of Hamiltonian PDE. In particular, we discuss the
theory of affine classical W-algebras and apply it to construct a large class
of integrable Hamiltonian PDE. We also discuss non commutative Hamiltonian PDE
in the framework of double PVA, and differential-difference Hamiltonian
equations in the framework of multiplicative PVA.
