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A microscopic approach is presented for calculating general properties of interacting Brownian particles under steady shearing. We start from exact expressions for shear-dependent steady-state averages, such as correlation and structure functions, in the form of generalized Green-Kubo relations. To these we apply approximations inspired by the mode coupling theory (MCT) for the quiescent system, accessing steady-state properties by integration through the transient dynamics after startup of steady shear. Exact equations of motion, with memory effects, for the required transient density correlation functions are derived next; these can also be approximated within an MCT-like approach. This results in closed equations for the nonequilibrium stationary state of sheared dense colloidal dispersions, with the equilibrium structure factor of the unsheared system as the only input. In three dimensions, these equations currently require further approximation prior to numerical solution. However, some universal aspects can be analyzed exactly, including the discontinuous onset of a yield stress at the ideal glass transition predicted by MCT. Using these methods we additionally discuss the distorted microstructure of a sheared hard-sphere colloid near the glass transition, and consider how this relates to the shear stress. Time-dependent fluctuations around the stationary state are then approximated and compared to data from experiment and simulation; the correlators for yielding glassy states obey a "time-shear-superposition" principle. The work presented here fully develops an approach first outlined previously [Fuchs and Cates, Phys. Rev. Lett. 89, 248304 (2002)], while incorporating a significant technical change from that work in the choice of mode coupling approximation used, whose advantages are discussed.