X-ray absorption spectra have been measured for NiO, beta-Ni(OH)(2), alpha-Ni(OH)(2), LiNiO2, and KNiIO6 samples, which contain nickel with valency in the range 2-4. Information on the local structure and nature of bonding of nickel compounds has been derived using theoretical standards generated with the FEFF code. The Ni K-edge energy was found to shift to higher values by about 1.5 eV per unit change in valency of nickel. The energy of the preedge peak (generally attributed to the transition from the Is core states to the 3d unoccupied states) shifts to higher values by about 0.6 eV per unit change in valency of nickel. A many body amplitude reduction factor (S-0(2)) of 0.77 +/- 0.03 for Ni K-edge absorption can be used to scale theoretical spectra to fit the experimental ones in order to accurately determine the coordination numbers for compounds with complex structures. Our results show that chemical effects are very small and can be ignored for reliable structural analysis. Results of local structure for the first, second, and third coordination spheres for NiO are consistent with those derived from X-ray diffraction data. The results for beta-Ni(OH)(2) are much-closer in agreement with those derived from the neutron diffraction data rather than the X-ray diffraction data. The results for alpha-Ni(OH)(2) differ significantly from those based on the idealized structure model proposed by Bode et al. In fact, the structure within the hexagonal planes for alpha-Ni(OH)(2) is similar to that for beta-Ni(OH)(2). The apparent contraction in the Ni-Ni distance of the third coordination sphere for alpha-Ni(OH)(2) relative to that for beta-Ni(OH)(2), which was previously reported by others, is not real and can be attributed to structural disorder. The Ni-O bond length for divalent nickel with an octahedral coordination is in the range 2.06-2.08 Angstrom. The first coordination sphere for LiNiO2 consists of four and two oxygens at 1.91 and 2.06 Angstrom, respectively, rather than six oxygens at 2.04 Angstrom, as predicted by the X-ray diffraction data. The short distance is characteristic of Ni3+-O2- bonds and Ni2+-O- bonds, while the long distance is characteristic of Ni2+-O2- bonds. The local structure for LiNiO2 is consistent with the fact that the charge-compensating mechanism is largely due to oxygen 2p holes. The Ni-Ni second-shell distance for LiNiO2 is consistent with that based on the X-ray and neutron diffraction data. The Ni-O bond length for quadrivalent Ni in KNiIO6 is 1.88 Angstrom. The Ni-O bond length varies with nickel valency in a nonlinear manner.