We study the development and evolution of buoyant river plumes on the continental shelf. Our calculations are based on three-dimensional numerical simulations, where the river runoff is introduced as a volume of zero salinity water in the continuity equation and mixing is provided by the model's turbulence closure scheme and wind forcing. In the absence of wind forcing the modeled river plumes typically consist of an offshore bulge and a coastal current in the direction of Kelvin wave propagation. We propose a plume classification scheme based on a bulk Richardson number, which expresses the relative magnitude of the buoyancy-induced stratification versus the available mixing. When the ratio of the discharge and shear velocities is greater (less) than 1, the plume is categorized as supercritical (subcritical); that is, the width of the bulge is greater (less) than the width of the coastal current. Supercritical plumes are often characterized by a meandering pattern along the coastal current, caused by a baroclinic instability process. For a given discharge, subcritical plumes are produced by large mixing and/or shallow water depths. In the presence of wind forcing the favorable conditions for offshore removal of coastal low-salinity waters include high river runoff and strong upwelling-favorable wind stress. When the rivers are treated as individual sources of freshwater ("point source" behavior), the wind-driven flow may exhibit substantial spatial variability. Under the above removal conditions, strong offshore transport takes place in "jetlike" flow regions within the river plume, in contrast to the downwind acceleration of adjacent waters. When the rivers are treated as a long "line source" of freshwater, the plume region resembles a coastal low-salinity band and the above removal conditions trigger offshore transport that is most pronounced at the "head" of the source.