This book studies the motion of suspensions, that is, of mixtures of a viscous incompressible fluid with small solid particles that can interact with each other through forces of non-hydrodynamic origin. In view of the complexity of the original (microscopic) system of equations that describe such phenomena, which appear both in nature and in engineering processes, the problem is reduced to a macroscopic description of the motion of mixtures as an effective continuous medium. The focus is on developing mathematical methods for constructing such homogenized models for the motion of suspensions with an arbitrary distribution of solid particles in a fluid. In particular, the results presented establish that depending on the concentration of the solid phase of the mixture, the motion of suspensions can occur in two qualitatively different modes: that of frozen or of filtering particles. Being one of the first mathematically rigorous treatises on suspensions from the viewpoint of homogenization theory, this book will be useful to graduate students and researchers in applied analysis and partial differential equations as well as to physicists and engineers interested in the theory of complex fluids with microstructure.