The primary concern of this thesis is to develop inversion methods which may be applied to observations of seismic anisotropy. I achieve this through the implementation of a non-linear global optimization scheme known as a Genetic Algorithm. I apply this technique to the inversion of quasi-compressional velocities obtained from laboratory measurements and shear-wave splitting observations measured from an azimuthal VSP experiment and two near-offset VSPs. These VSP experiments were conducted at the Conoco Borehold Test Facility, Oklahoma which presents an ideal setting due to its geological and structural simplicity and the availability of a-priori information, for example from borehole televiewer images, core samples and logging information. I compare the information derived from these sources with the results obtained from the inversion of shear-wave birefringence measurements. Comparative studies of this type are important in establishing the relation between the observed seismic anisotropy and equivalent media systems due to such phenomena as aligned fractures or fine layering. The inversion results for the azimuthal VSP and near-offset VSPs are in good agreement with each other and also with the a-priori data. The azimuthal VSP inversion allowed the identification of a significant feature associated with shear-wave propagation in anisotropic media known as a shear-wave singularity. Another significant inversion result is the suggestion that the observed seismic anisotropy is due to a sub-vertical fracture system. This result is supported through the analysis of shear-wave observations obtained from the near-offset CSP experiments. I suggest that fracture dip may be determined using appropriate acquisition geometries such as an opposite azimuth VSP for which two VSPs are recorded with sources located diametrically opposite each other about the well. A significant correlation is observed to exist between the lithology at the Conoco Borehole Test Facility and the degree of shear-wave birefringence. Specifically it appears that the shear-wave birefringence is most pronounced for wave propagation through the sandstone formations which are known to be heavily fractured. Correlations of this type are important if any confidence is to be placed in the interpretation of shear-wave birefringence in terms of fracture systems.