Relative to its extremely low abundance in the Earth's crust, tellurium is the most mineralogically diverse chemical element, with over 160 mineral species known that contain essential Te, many of them with unique crystal structures. We review the crystal structures of 703 tellurium oxysalts for which good refinements exist, including 55 that are known to occur as minerals. The dataset is restricted to compounds where oxygen is the only ligand that is strongly bound to Te, but most of the Periodic Table is represented in the compounds that are reviewed. The dataset contains 375 structures that contain only Te4+ cations and 302 with only Te6+, with 26 of the compounds containing Te in both valence states. Te6+ was almost exclusively in rather regular octahedral coordination by oxygen ligands, with only two instances each of 4- and 5-coordination. Conversely, the lone-pair cation Te4+ displayed irregular coordination, with a broad range of coordination numbers and bond distances. A threshold was applied for Te4+–O links of ∼2.45 Å or 0.3 valence units with some flexibility, as a criterion to define strongly bound Te–O polymers and larger structural units. Using this criterion, Te4+ cations display one-sided 3-, 4- or 5-coordination by oxygen (with rare examples of coordination numbers 2 and 6). For both valence states of Te, examples are known of TemOn complexes which are monomeric (m = 1; neso), noncyclic finite oligomers (soro), rings (cyclo), infinite chains (ino), layers (phyllo) and frameworks (tecto tellurates). There is a clear analogy to the polymerization classes that are known for silicate anions, but the behaviour of Te is much richer than that of Si for several reasons: (1) the existence of two cationic valence states for Te; (2) the occurrence of multiple coordination numbers; (3) the possibility of edge-sharing by TeOn polyhedra; (4) the possibility for oxygen ligands to be 3-coordinated by Te; and (5) the occurrence of TemOn polymers that are cationic, as well as neutral or anionic. While most compounds contain only one or two symmetrically distinct types of Te atom, Pauling's Fifth Rule is frequently violated, and stoichiometrically simple compounds such as CaTeO3 can have polymorphs with up to 18 distinct Te sites. There is a tendency for local symmetry features such as the threefold axis of a TeO6 octahedron or the acentric symmetry of a Te4+On polyhedron to be inherited by the host structure; the latter in particular can lead to useful physical properties such as nonlinear optical behaviour. We develop for the first time a hierarchical taxonomy of Te-oxysalt structures, based upon (1) valence state of Te; (2) polymerization state of TemOn complexes; (3) polymerization state of larger strongly-bound structural units that include non-Te cations. Structures are readily located and compared within this classification.