  We develop the formalism of double Poisson vertex algebras (local and
non-local) aimed at the study of non-commutative Hamiltionan PDEs. This is a
generalization of the theory of double Poisson algebras, developed by Van den
Bergh, which is used in the study of Hamiltonian ODEs. We apply our theory of
double Poisson vertex algebras to non-commutative KP and Gelfand-Dickey
hierarchies. We also construct the related non-commutative de Rham and
variational complexes.
