We have presented the basic theory and examples of an algorithm that performs anisotropic traveltime tomography. The algorithm generalizes the well known techniques of tomographic traveltime inversion by using models discretized into a set of homogeneous, elliptically-anisotropic cells, where each cell is characterized by two slownesses, one vertical and the other horizontal. Both components of the slowness can be estimated simultaneously provided that the range of ray angles is great enough. Otherwise the problem becomes ill-conditioned. As expected for cross-well geometries, the ill-conditioning is more severe in the estimation of the vertical component of the slowness. This is another reason why we should combine information from different geometries (VSP, surface-to-surface, and cross-well), since taken alone they may have limited aperture.