A brief account is given of the spectrum of extended-object solutions to supergravity theories that saturate Bogomol’ny-Prasad-Sommerfield bounds, and which therefore are candidates for exact states in the underlying string theory. These are organized into a coherent scheme by a pattern of vertical and diagonal dimensional reduction trajectories on a plot of worldvolume versus spacetime dimension. These intersecting reduction trajectories are related by duality symmetries at the intersection points, and these duality symmetries govern also the multiplicities of the p-branes at each (worldvolume, spacetime) dimension.