The design of supramolecular hosts for anions began with simple diaza bicycles, named katapinands, and has evolved over the last 40 years to a number of elegantly designed receptors capable of binding many different anions. About the same time the term cryptand appeared in reference to another bicyclic compound that was selective for alkaline-earth ions. Since the first report these simple bicycles, a vast arena of hosts has appeared, including acyclic, monocyclic, and other multicyclic supramolecular receptors. Studies of these systems have revealed considerable information about anion coordination chemistry, including the fact that many of these complexes mimic their transition-metal corollaries in terms of coordination numbers. However, for anions interactions occur via H-bonding most often, rather than the coordinate covalent or dative bonds observed in transition-metal coordination. This critical review examines the design of enclosed, primarily bicyclic cryptands as hosts for anions, with a small scattering of higher polyhedra when deemed appropriate to the discussion. In order to show the development (evolution) of the field, key examples of early work will be noted and compared with more recent developments (136 references).