We extend seismic source theory to general anisotropic media for the numerical evaluation of spectral amplitudes of point sources in multilayered anisotropic crustal models. First, we obtain an explicit representation of the spectral elastodynamic Green's tensor in general homogeneous anisotropic media as a sum of three integrals over the corresponding three slowness surfaces. The multidimensional stationary phase principle is then applied to derive an asymptotic approximation at the far field. In the case of azimuthally isotropic media, we offer an alternative representation of the Green's tensor and its ensuing displacements fields in the form of an exact Hankel transform over the horizontal wave number variable. The total field is specified here in terms of two potentials; an SH potential and a mixed quasi-transverse/quasi-longitudinal potential, both of which assume the role of two scalar Green's functions. The availability of the Green's tensor in analytical form enables one to obtain readily numerical solutions for a wide selection of media and sources. It is shown that the radiation field of an explosion has the following new features: (1) Quasi-transverse waves are created with four and eight lobe patterns; (2) quasi-longitudinal waves are generated for the collatitudinal displacement with four lobe patterns; (3) the energy ratio SV/P may reach the value of 20 for more than 50% of the azimuths in crustal structures such as tuff and shales; and (4) radiation patterns for vertical shear waves are created which are indistinguishable from corresponding waves produced by earthquake faults. Our formalism allows us to establish a Haskell-type matrix algorithm for a multilayered azimuthally isotropic half-space, which enables us to calculate body waves and surface waves in real-earth crustal models. ¿ American Geophysical Union 1990