Work carried out during the last few years on the geometry of Hamiltonian systems, including hydrodynamic-type systems and their Poisson brackets, is reviewed. The theory of the integrability of hydrodynamic- type systems is presented. Applications of the formalism developed to problems of the hydrodynamics of weakly deformed soliton lattices - slow modulations of families of periodic and quasiperiodic solutions of nonlinear evolution systems, are described.