The molecular and electronic structures of transition metal complexes, [MOCl5]n- (n = 2 for M = V,Nb,Ta and n = 1 for Mo,W) and mixed-metal polyoxometalates, [M'M5O19]3-V,Nb,Ta, M = Mo,W) containing a single terminal oxo group on each metal, and of complexes of the uranyl ion [UO2]2+, [UO2(H2O)5]2+ and [UO2Cl4]2-, have been calculated using density functional methods. The calculated structures of the complexes are in good agreement with available experimental parameters. For the mixed-metal hexametalates, for which no crystallographic data is available, the calculations predict a small tetragonal compression of the clusters with only minor structural changes compared to the parent molybdate and tungstate. The metal oxygen bonding in these anions has been probed using Mayer-Mulliken, bond energy and atoms in molecule analyses (AIM). These methods provide a consistent description of the bonding in polyoxometalates. The terminal bonds between transition metal or uranium and oxygen atoms have large sigma and pi components with the pi contributions exceeding the sigma bonding. The transition metals utilize their d orbitals almost exclusively to bond to oxygen whilst uranium uses both its 5f and 6d orbitals. Oxygen atom charges increase and covalency indexes decrease with coordination number, with a marked separation of these terms according to the oxygen atom type. The total valency and AIM energies of the oxygen atoms are predicted to be almost constant for all types of oxygen site. The constancy of the bonding power of the oxygen atoms appears to be an important factor in determining the gross structures and details of the bonding in polyoxometalates. The Mayer Mulliken approach provides direct characterization of the bonding power of atoms and the extent of the interaction between pairs of atoms that is consistent with the results of the considerably more computationally demanding bond energy and AIM approaches.