We present a generic lagrangian, in arbitrary spacetime dimension D, describing the interaction of a dilaton, a graviton and an antisymmetric tensor of arbitrary rank d. For each D and d, we find black p-brane solutions where p = d - 1 and d = D - d - 2. These solutions display a spacetime singularity surrounded by an event horizon and are characterized by a mass per unit p-volume, M(d), and a topological '' magnetic'' charge g(d), obeying square-root 2 kappa M(d) greater-than-or-equal-to g(d). The theory also admits elementary p-brane solutions with ''electric'' Noether charge ed, obeying the Dirac quantization rule e(d) g(d) = 2pin, n = integer. We then present the lagrangian describing the theory dual to the original theory, whose antisymmetric tensor has rank d and for which the roles of topological and elementary solutions are interchanged. In the extreme limits square-root 2 kappa M(d) = g(d) or square-root 2 kappa M(d) = e(d), the singularity and event horizon coalesce. In this case, the metrics and their duals are mutually nonsingular. For specific values of D and d, these extreme solutions also exhibit supersymmetry and some may be identified with previously classified heterotic, Type IIA and Type IIB super p-branes. In particular, within the context of Type II theory, the electric/magnetic duality of Gibbons and Perry in D = 4 is seen to be a consequence of particle/sixbrane duality in D = 10. Among the new solutions is a self-dual superstring in D = 6.
