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I show how to compute the properties of a class of charged black holes in anti-de Sitter space-time, in diverse dimensions. These black holes are shown to be solutions of consistent Einstein-Maxwell truncations of gauged supergravities, which are shown to arise from the inclusion of rotation in the transverse space. Rich thermodynamic phase structures for these systems are uncovered, which display classic critical phenomena, including structures isomorphic to the van der Waals-Maxwell liquid-gas system. In that case, the phases are controlled by the universal `cusp' and `swallowtail' shapes familiar from catastrophe theory. All of the thermodynamics is consistent with field theory interpretations via holography, where the dual field theories can sometimes be found on the world volumes of coincident rotating branes.