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A 'viscoelastic' fluid is material which has both the viscous character of a Newtonian fluid and the ability to store energy that characterizes an elastic solid. Such fluids are quite common both in nature and in industrial applications, some common exa mples being biological fluids, many foods, polymer solutions and melts, and most colloidal suspensions and emulsions. In our group we are studying how small particles move and interact with each other when suspended in these complex fluids. There is amp le experimental evidence that the behavior of particles in viscoelastic fluids differs dramatically from that in Newtonian fluids: Sedimentation in bulk suspensions is accompanied by the formation of large-scale heterogeneities in particle concentration; small clusters of sedimenting spheres form vertical chains and fall in rows; and spheres sedimenting near a wall are attracted to the wall, while spheres in pressure-driven flow through a channel move away from the wall and toward the channel center. We are performing simulations and sedimentation experiments in order to understand these and other phenomena. In our most recent simulations, the 'viscoelastic fluid' is represented as a suspension of bead-and-spring dumbbells in a Newtonian solvent. The dumbbells impart to the medium a 'finite memory' and the ability to store and release energy. These dumbbell suspensions show negligible shear-thinning, and hence fall into the class of constant-viscosity, elastic fluids known as Boger fluids. Results w ill be presented for sedimentation of one and two spheres and for sedimentation of non-spherical particles in such a medium. Particle motion near walls, both in stagnant fluids and pressure-driven flows, will also be discussed. Our simulation results sh ow good qualitative agreement with experimental observations, and provide insight into why particles in complex fluids behave as they do.