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Particle-particle interactions in viscoelastic fluids differ qualitatively from those in Newtonian fluids. Even when the fluids are only weakly viscoelastic, suspension microstructures can be strongly altered over time scales of interest. At low Deborah numbers, we have explored the nature of these interactions in detail, and analytic expressions for multiparticle interactions have been derived for dilute suspensions. Dynamic simulations performed with these expressions show that particles sedimenting in viscoelastic fluids move together and align in the direction of gravity, and neutrally buoyant particles in shear flows align in the direction of the velocity. Both of these predictions are in qualitative agreement with experimental observations. In an unbounded, sedimenting suspension, the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. A particle conservation equation that accounts for these two influences shows that the suspension microstructure can become unstable to concentration fluctuations in the direction perpendicular to gravity. As a result, vertical, particle-rich columns form, along with vertical particle-depleted regions. Theoretical predictions of the length scale of these columns and the time scale for their formation are in agreement with experimental data. One consequence of this instability is that, in viscoelastic fluids, concentrated suspensions may sediment more rapidly than dilute suspensions.