The geodesic law for test particles is one of the fundamental principles of general relativity and is extensively used. It is thought to be a consequence of the field laws but no rigorous proof exists. This thesis is concerned with a precise formulation of the geodesic law for test particles and with the extent of its validity. It will be shown to be true in certain cases but not in others. A rigorous version of the Infeld/Schild theorem is presented. Several explicit examples of both geodesic and non-geodesic motion of singularities are given. In the case of a test particle derived from a test body with a regular internal stress-energy tensor, a proof of the geodesic law for an ideal fluid test particle under plausible, explicitly stated conditions is given. It is also shown that the geodesic law is not generally true, even for weak fields and slow motion, unless the stress-energy tensor satisfies certain conditions. An explicit example using post-Newtonian theory is given showing how the geodesic law can be violated if these conditions are not satisfied. / Thesis (Ph.D.)-University of Natal, Durban, 1998.