The problem of inverse ray tracing in a homogeneous anisotropic elastic solid is considered, with specific application to crack sizing. The data is assumed to be in the form of travel times of diffracted ultrasonic signals between transducers positioned on an exterior surface of the body. Both pulse-echo and pitch-catch data are considered. First, it is assumed that the wave speeds are unknown and must be obtained as part of the inversion procedure. The specific problem of locating a crack tip in a two-dimensional geometry is investigated. It is found that travel time data on the exterior surface suffices to locate the crack tip only if the material is isotropic. If the material is anisotropic, we must be able to move the source and/or receiver in the direction normal to the surface. The same problem is considered with the source and receiver positioned in a surrounding isotropic material, e.g., a water bath. It is shown that the ray inversion is now possible only if the solid is isotropic, the problem being underdetermined for an anisotropic solid. Numerical results are presented for a synthetic experiment in which a finite crack is present in some anisotropic elastic solids. Next, the problem is considered when the speeds are known a-priori. It is shown that a crack edge can be mapped by a local approximation procedure.