We study diffusion in a suspension of hard spheres with neglect of hydrodynamic interactions. The wavenumber-dependent long-time collective- and self-diffusion coefficients are calculated for a wide range of densities. At low density we obtain exact analytic results. We use a cluster expansion to express the diffusion coefficients in terms of evolution operators of progressively increasing complexity describing diffusion of 2,…,s interacting particles. At low density it suffices to solve the dynamical two-body problem. We assume that at intermediate and high density the main contribution comes from the two-body term, at least for not too small values of the wavenumber. We evaluate the diffusion coefficients in this two-body diffusion approximation. In the calculation the equilibrium pair correlation function is needed. At intermediate density we employ the first few terms in the virial expansion of the equilibrium pair correlation function. At high density we use the closed form of expression for the structure factor found in the mean spherical approximation. The collective-diffusion coefficient shows a variation with a wavenumber similar to the equilibrium structure factor, where-as the self-diffusion coefficient has smooth monotonic behaviour.