Suspensions of solid particles, drops and bubbles in liquids occur frequently in a variety of industrial processes and in nature. Hence their behavior and modeling in flow situations has attracted a great deal of attention from both the practical and fundamental points of view. Until recently, such systems were studied either experimentally or via the solution of model equations, the reliability of which was often subject to question. Recently, however, direct numerical simulations (DNS), where the detailed flow field around each dispersed phase (or particle) is obtained by the numerical solution of the fundamental governing equations, are making it possible to examine aspects of the flow structures previously unattainable. Although such simulations make it possible to examine the unsteady flow of hundreds of particles, predictions of the behavior and properties of large-scale natural and engineered systems will continue to rely on model equations that characterize the overall average flow field. The major role of direct numerical simulations will be to provide insight and quantitative data to close the model equations. The closure, more precisely modeling the effect of unresolved scales with those that are resolved, has relied mostly on experimental data and simple scaling arguments. The success of numerical simulations and the abundance of very detailed data, have now created considerable urgency to access the state of the art modeling and the feasibility of the use of the results of simulations in developing closures.