In global ocean dynamics Rossby waves play a vital rôle in the long-term distribution of vorticity; knowledge of the interaction between these waves and topography is crucial to a full understanding of this process, and hence to the transportation of energy, mixing and ocean circulation. The interaction of baroclinic Rossby waves with abrupt topography is the focus of this study. In this paper we model the ocean as a continuously stratified fluid for which the linear theory predicts a qualitatively different structure for the wave modes than that predicted by barotropic or simple layered models, even if most of the density variation is confined to the thermocline. We consider the scattering of a westward-propagating baroclinic Rossby wave by a narrow ridge on the ocean floor, modelled by a line barrier of infinite extent, orientated at an arbitrary angle to the incident wave. Transmission and reflection coefficients for the propagating modes are found using both an algebraic method and, in the case where this breaks down, matched asymptotic expansions. The results are compared with recent analyses of satellite altimetry data.