The diffusion coefficients of polystyrene latex spheres in both Newtonian and elastic liquids have been measured using dynamic light scattering. The diffusion coefficients of the latex particles measured in glycerol/water (Newtonian) solutions obey Stokes-Einstein behavior over a range of solvent viscosity and temperature. Two apparent diffusion coefficients for the particles are measured in viscoelastic polyacrylamide solutions and are designated D-fast and D-slow. The apparent fast diffusion coefficients measured in the polyacrylamide solutions show an increase to a maximum, above that measured in the solvent water, with increasing polyacrylamide concentration. The apparent fast diffusion coefficient is seen to increase to twice that measured in the water solvent. At higher polyacrylamide concentrations the observed D-fast values decrease below the value obtained in the solvent water. D-fast increases with the scattering vector (q) while D-slow is independent of q.