We report on a joint experimental-theoretical study of diffusion processes in dense suspensions of charge-stabilized colloidal spheres. Numerous transport properties have been calculated by means of a recently developed accelerated Stokesian dynamics simulation tool, and using methods of many-body theory adapted to colloidal soft matter systems. The simulation and theoretical methods describe quantitatively our static and dynamic synchrotron radiation scattering experiments on colloidal particles, globular proteins and DNA fragments. Salient results of this study comprise the absence of screening of hydrodynamic flow in salt-free suspensions, and a hydrodynamic explanation for the unexpected non-monotonic concentration dependence of self-diffusion in suspensions of moderately charged particles.