The barotropic shear layer in a rotating fluid is studied in a laboratory experiment. Through the rotation of circular sections in the base and lid of a circular tank relative to a background rotation of the entire system, a vertical layer of strong horizontal shear develops, the Stewartson layer. Above a critical shear, the shear layer breaks up through barotropic instability, which is an inertial instability The flow then develops a string of vortices along the shear zone. It will be shown that the transition from an axisymmetric flow to regular vortices occurs through a Hopf bifurcation. Subsequent transitions to more complex flows, such as modulated vortices, chaos and highly irregular flow, will be presented briefly, while the main points of this paper are the primary instability, steady vortices and their nonlinear dynamics. Among the issues discussed is the sensitivity of the flow to the direction of the differential shear. The experimental data will be used to test the ability of boundary layer theory and quasi-geostrophic theory to predict the onset of instability and the range of unstable wavenumbers.