The structure and bonding in oxidized and reduced decatungstate anions have been investigated using density-functional methods. The computational-experimental agreement is good for the geometrical parameters of the oxidized species, The electronic structure of the anions has been probed with molecular-orbital, Mulliken-Mayer, and bonding-energy approaches, and the various analyses are in general accordance with spectroscopic evidence and theoretical models. The results have indicated that W-O interactions are largely W d-O p in character, and that sigma and pi bonds link the metal centers to terminal and bridging (O-2c) oxygen atoms. Some W-O-2, orbital interactions can be represented as [W4O4] or [W6O6] closed-loop structures, but these bonding modes have not been found to make a particularly outstanding contribution to the stability of the molecules. Mayer indexes correspond to (fractional) multiple, approximately single, and low-order character for terminal, bridging, and internal bonds, respectively, and the valency analysis has yielded similar bonding capacities for the different oxygen atoms. A distribution of the negative charge over all types of oxygen sites, and metal charges and orbital populations considerably different from the formal assignments have been obtained from the Mulliken analysis. Minor structural changes have been detected in reduced decatungstates, in accord with the general properties of the orbitals occupied by the added electrons.