Nonequilibrium Brownian dynamics simulations are used to study the effect of polydispersity on the thermodynamics, rheology, microstructure, and shear-induced disorder-order transition in suspensions of charged colloids. Approximately 43 000 particles with 2000 different components of a discretized Schulz distribution at polydispersities from 0% to 30% are simulated on a massively parallel computer. Recent advances in the integral equation theory for polydisperse suspensions are tested and verified with respect to both structure and equilibrium mechanical properties. The low shear rate theology for both monodisperse and polydisperse suspensions is found to be well represented by the Ree-Eyring model. At higher shear rates an ordered "string" phase is shear induced for low polydispersities (< 10%). Increasing the polydispersity further (> 20%) inhibits the ordering, suggesting the existence of a critical polydispersity beyond which a colloidal suspension cannot be induced into an ordered state by shearing; The validity of a generalized, nonequilibrium Stokes-Einstein relationship for polydisperse colloids is also investigated.