The amplitude variation with offset (AVO) responses of elastic transversely isotropic media are sensitive to contrasts in both of Thomsen's anisotropic parameters delta and epsilon . The equation describing P-P reflections indicates that the smaller the contrasts in isotropic properties (compressional velocity, shear velocity, and density) and the larger the contrasts in delta and epsilon across an interface of reflection, the greater the effects of anisotropy on the AVO signature. Contrasts in delta are most important under small-to-medium angles of incidence as previously described by Banik (1987), while contrasts in epsilon can have a strong influence on amplitudes for the larger angles of incidence commonly encountered in exploration (20 degrees and beyond). Consequently, using Rutherford and Williams' AVO classification of isotropic gas sands, type I gas sands overlain by a transversely isotropic (TI) shale exhibit a larger decrease in AVO than if the shale had been isotropic, and type III gas sands overlain by a transversely isotropic (TI) shale exhibit a larger increase in AVO than if the shale had been isotropic. Furthermore, it is possible for a 'type III' isotropic water sand to exhibit an 'unexpected' increase in AVO if the overlying shale is sufficiently anisotropic.More quantitative AVO interpretations in TI media require considerations of viscoelastic TI in addition to elastic TI and lead to complicated integrated earth models. However, when elastic and viscoelastic TI have the same axis of symmetry in a shale overlying an isotropic sand, both elastic and viscoelastic TI affect the overall AVO response in the same direction by constructively increasing/decreasing the isotropic component of the AVO response. Continued efforts in this area will lead to more realistic reservoir models and hopefully answer some of the unexplained pitfalls in AVO interpretation.