Black hole solutions of supergravity theories form families that realizing the deep nonlinear duality symmetries of these theories. They form orbits under the action of these symmetry groups, with extremal (i.e. BPS) solutions at the limits of such orbits. An important technique in the analysis of such solution families employs timelike dimensional reduction and exchanges the stationary black-hole problem for a nonlinear sigma-model problem. Families of extremal or BPS solutions are characterized by nilpotent orbits under the duality symmetries, based upon a tri-graded or penta-graded decomposition of the corresponding duality-group algebra.