In isotropic anelastic media, the phase velocity of an inhomogeneous plane body wave, which is a function of Q and the degree of inhomogeneity &ggr;, is significantly less than the corresponding homogeneous wave phase velocity typically only if &ggr; is very large (unless Q is unusually low). Here we investigate inhomogeneous waves in anisotropic anelastic media, where phase velocities are also functions of the direction of phase propagation &thgr;, and find that (1) the low phase velocities can occur at values of &ggr; which are substantially less than the isotropic values and that they occur over a limited range of oblique directions &thgr;, and (2) for large positive valus of &ggr;, there are ranges of oblique directions &thgr; in which the inhomogeneous waves cannot propagate at all because there is no physically acceptable solution to the dispersion relation. We show examples of how the waves of case 1 can occur in practice and cause a number of anomalous wave propagation effects. The waves of case 2, though, do not arise in practice (they do not correspond to any points on the horizontal slowness plane). We also show that in the decomposition of a cylindrical wave into plane waves, inhomogeneous plane waves occur whose amplitudes grow in the direction of phase propagation and that this direction is away from the receiver to which they are contributing. The energy in these waves does, however, travel toward the receiver, and their amplitudes decay in the direction of energy propagation. We also show that if the commonly used definition for the quality factor in an isotropic medium, Q=-Re(μ)/Im(μ) where μ is a complex modulus, is applied to an anisotropic anelastic medium in order to study absorption anisotropy, a generally unreliable measure of the anelasticity of inhomogeneous wave propagation in a given arbitrary direction is obtained. The more fundamental definition based on energy loss (i.e., 2&pgr;/Q=ΔE/E) should be used in general, and we present some basic formulas for this quantity, as well as others, for plane waves in transversely isotropic anelastic media. ¿ American Geophysical Union 1994