U duality and central charges in various dimensions revisited

A geometric formulation which describes extended supergravities in any dimension in the presence of electric and magnetic sources is presented. In this framework, the underlying duality symmetries of the theories are manifest. Particular emphasis is given to the construction of central and matter charges and to the symplectic structure of all D = 4, N-extended theories. The latter may be traced back to the existence, for N > 2, of a flat symplectic bundle which is the N > 2 generalization of N = 2 Special Geometry