Two models, one numerical and one analytical, are used to investigate mechanisms for the generation and flow of the Leeuwin Current observed off the west coast of Australia. Three numerical experiments are conducted using the Bryan–Cox Ocean General Circulation Model. In the first experiment an alongshore density gradient is imposed, while in the latter two experiments the additional effects of warm, fresh North West Shelf waters are considered. The alongshore density gradient in the Indian Ocean produces an onshore geostrophic flow which turn southward and intensifies as it flows along the coast of Western Australia and into the Great Australian Bight. Maximum alongshore surface velocities occur just off the shelf break and below this poleward current is a weak equatorward flow. The magnitudes of the velocity vectors and associated advection of temperature and salinity, and the width, structure and geographical location of the current all agree well with field observations. The effects of the warm, fresh North West Shelf waters are to enhance the alongshore barotropic current on the shelf. This response is mainly local and diminishes with alongshore distance. Model results suggest that the Leeuwin Current a baroclinic current driven by an alongshore density gradient in the Indian Ocean, maintained by equatorward surface heating and poleward surface cooling. The numerical results further predict the existence of a cyclonic circulation off Northwest Cape in the top 120 m of water. It is deduced that the seasonal modification of the Leeuwin Current is due to local winds and seasonal variation in the north–south density gradient. A linear analytical model of the continental shelf cicculation coupled to a two layer open ocean confirms the conclusions drawn from the numerical model, and shows that the shelf circulation obeys a simple balance between alongshore pressure gradient forces (due to steric height gradients), and bottom friction. It is shown that in the absence of vertical mixing, the presence of a sloping shelf is necessary for the existence of the trapped eastern boundary current. Without the sloping shelf the current radiates offshore through baroclinic Rossby wave propagation. For parameters applicable to the Leeuwin Current, the analytic model predicts realistic alongshore transports.