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Polyhedral molecules are appealing for their eye‐catching architecture and distinctive chemistry. Perfluorination of such, often greatly strained, compounds is a momentous challenge. It drastically changes the electron distribution, structure and properties. Notably, small high‐symmetry perfluoropolyhedranes feature a centrally located, star‐shaped low‐energy unoccupied molecular orbital that can host an extra electron within the polyhedral frame, thus producing a radical anion, without loss of symmetry. This predicted electron‐hosting capacity was definitively established for perfluorocubane, the first perfluorinated Platonic polyhedrane to be isolated pure. Hosting atoms, molecules, or ions in such "cage" structures is, however, all but forthright, if not illusionary, offering no easy access to supramolecular constructs. While adamantane and cubane have fostered numerous applications in materials science, medicine, and biology, specific uses for their perfluorinated counterparts remain to be established. Some aspects of highly fluorinated carbon allotropes, such as fullerenes and graphite, are briefly mentioned for context. [ABSTRACT FROM AUTHOR]