Suspensions of monodisperse spheres dispersed in a liquid polymer have been studied in shear flow. The particle diameters range from 0.18 to 2.7 mum. The measured properties include linear dynamic moduli, steady-state shear viscosities, and nonlinear stress relaxation moduli. At small volume fractions and/or large particle sizes, the linear and nonlinear behavior can be explained on the basis of hydrodynamic effects caused by the presence of the particles. In that case, the nonlinear relaxation moduli display time-strain separability. The concentration dependence of the damping function at large strains is similar to that of the linear properties. At small interparticle distances, a weak particulate structure develops that mainly alters the low shear viscosities and the low-frequency moduli. The effect of this structure on the stresses can be gradually eliminated by applying larger strains, as demonstrated by the strain dependence of the nonlinear relaxation moduli in step-strain experiments.