Four theoretical studies on anharmonic thermal vibration by Willis (1969), Matsubara (1975 a, b), Kurki-Suonio, Merisalo and Peltonen (1979), and Johnson (1969) are reviewed and compared with each other. Generalized Willis method by Tanaka and Marumo (1983) is also described. Experimental works are summarized to display the charcteristics of each method. Anharmonic thermal vibration of the Cu atoms in KCuF₃ crystals and the O atoms in α-Al₂O₃ crystals were studied by the authors. The study on KCuF₃ crystals exhibited that the aspherical electron distribution and thermal vibration, which appeared on the deformation density map as superposed peaks, could be seperable. Anharmonic thermal vibration of the O atoms in α-Al₂O₃ crystals suggested possible diffusion paths of the O atoms. The maxima of potential along the paths are a little lower than the observed activation energy of the self-diffusion process of the O atoms.