Over the past four decades, the number of inorganic oxide and oxy-salt phases containing stoichiometric quantities of hexavalent uranium has increased exponentially from a few dozen to well over 700, and these structures have become well-known for their remarkable compositional diversity and topological variability. Considering the entirety of these compounds (i.e., crystal structures, conditions of synthesis, and geological occurrences) offers significant insight into the behavior of uranium in the solid state and in the nascent (typically aqueous) fluids. The structure hierarchy approach adopted here aims specifically to facilitate the recognition of useful patterns in the crystal-chemical behavior of hexavalent uranium (U6+). This work represents the third attempt at a structure hierarchy of U6+ compounds, with the first two being those of Burns et al. (1996) and Burns (2005), which considered 180 and 368 structures, respectively. The current work is expanded to include the structures of 727 known, well-refined synthetic compounds (610) and minerals (117) that contain stoichiometric quantities of U6+. As in the previous works, structures are systematically ordered on the basis of topological similarity, as defined predominantly by the polymerization of high-valence cations. The updated breakdown is as follows: (1) isolated polyhedra (24 compounds/0 minerals); (2) finite clusters (70 compounds/10 minerals); (3) infinite chains (94 compounds/15 minerals); (4) infinite sheets (353 compounds/79 minerals); and (5) frameworks (186 compounds/13 minerals). Within each of these major categories, structures are sub-divided on the basis of increasing connectivity of uranium (nearly always uranyl) polyhedra. In addition to elucidating common trends in U6+ crystal chemistry, this structure hierarchy will serve as a comprehensive introduction for those not yet fluent in the domain of uranium mineralogy and inorganic, synthetic uranium chemistry.