Research concerning the crystal chemistry of hexavalent U by the Environmental Mineralogy and Crystal Structures research group at Notre Dame has resulted in the description of more than 110 new structures of uranyl compounds (including 36 minerals). New insights into the crystal chemistry of U6+ are presented, with emphasis on recently discovered novel structural connectivities. The structural hierarchy of uranyl minerals and compounds, which was first established for 180 structures in 1996, has been extended to include 145 new structures. The hierarchy is based upon polymerization of polyhedra containing higher-valence cations, and consists of five distinct classes: structures containing isolated polyhedra (7), finite clusters of polyhedra (41), chains of polyhedra (52), sheets of polyhedra (184), and frameworks of polyhedra (41). The dominance of sheets in uranyl compounds (57% of known structures) arises from the unequal distribution of bond-valences within the uranyl polyhedra. Topological relations of the sheets in uranyl compounds are best understood by analysis of the topological distribution of anions within sheets in which sharing of polyhedral edges dominates, and by graphical representation of the connectivity of polyhedra in cases where sharing of vertices of polyhedra dominates the sheet.