We review the effects of molecular crowding on solute diffusion in solution and in cellular aqueous compartments and membranes. Anomalous diffusion, in which mean squared displacement does not increase linearly with time, is predicted in simulations of solute diffusion in media crowded with fixed or mobile obstacles, or when solute diffusion is restricted or accelerated by a variety of geometric or active transport processes. Experimental measurements of solute diffusion in solutions and cellular aqueous compartments, however, generally show Brownian diffusion. In cell membranes, there are examples of both Brownian and anomalous diffusion, with the latter likely produced by lipid-protein and protein-protein interactions. We conclude that the notion of universally anomalous diffusion in cells as a consequence of molecular crowding is not correct and that slowing of diffusion in cells is less marked than has been generally assumed.