The hydrodynamic transport properties of hard‐sphere dispersions are calculated for volume fractions (&fgr;) spanning the dilute limit up to the fluid–solid transition at &fgr;=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite‐size effects are derived. The effects of using various levels of approximation in computing both the far‐ and near‐field hydrodynamic interactions are also examined. The transport properties associated with freely mobile suspensions—sedimentation velocities, self‐diffusion coefficients, and effective viscosities—are determined here, while the corresponding properties of porous media are determined in a companion paper [Phys. Fluids31, xxxx (1988)]. Comparison of the simulation results is made with both experiment and theory. In particular, the short‐time self‐diffusion coefficients and the suspension viscosities are in excellent agreement with experiment.