We calculate the short-time translational and rotational self-diffusion coefficients of mixtures of colloidal hard spheres. The influence of hydrodynamic interactions is accounted for by a series expansion of the two-body mobility tensors, and by considering the leading hydrodynamic three-body term. Explicit results are presented for the rotational-translational diffusion coefficients, D-s,T(r) and D-s,T(t), of a dilute tracer component in a concentrated host dispersion of hard spheres. These coefficients are shown to depend strongly on the host volume fraction, phi, and on the size ratio, lambda, of tracer and host particles. For large lambda, the tracer diffusion coefficients are related to the viscosity of the host dispersion by effective Stokes-Einstein relations.