  In this paper the well-known Dubrovin-Novikov problem posed as long ago as
1984 in connection with the Hamiltonian theory of systems of hydrodynamic type,
namely, the classification problem for multidimensional Poisson brackets of
hydrodynamic type, is solved. In contrast to the one-dimensional case, in the
general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic
type cannot be reduced to a constant form by a local change of coordinates.
Generally speaking, such a Poisson bracket is generated by a nontrivial special
infinite-dimensional Lie algebra. In this paper we obtain a classification of
all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic
type for any number N of components and for any dimension n by
differential-geometric methods. A key role in the solution of this problem was
played by the theory of compatible metrics that had been earlier constructed by
the present author.
