  The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat
Hamiltonian partial differential equations. A PVA consists of a pair
$(\mathcal{A},\{\cdot_\lambda\cdot\})$ of a differential algebra $\mathcal{A}$
and a bilinear operation called the $\lambda$-bracket. We extend the definition
to the class of algebras $\mathcal{A}$ endowed with $d\geq1$ commuting
derivations. We call this structure a \emph{multidimensional PVA}: it is a
suitable setting to study Hamiltonian PDEs with $d$ spatial dimensions. We
apply this theory to the study of deformations of the Poisson brackets of
hydrodynamic type for $d=2$.
