Sedimentation and suspension flows play an important role in modern technology. This special issue joins nine recent contributions to the mathematics of these processes. The Guest Editors provide a concise account of the contributions to research in sedimentation and thickening that were made during the 20th century with a focus on the different steps of progress that were made in understanding batch sedimentation and continuous thickening processes in mineral processing. A major breakthrough was Kynch's kinematic sedimentation theory published in 1952. Mathematically, this theory gives rise to a nonlinear first-order scalar conservation law for the local solids concentration. Extensions of this theory to continuous sedimentation, flocculent and polydisperse suspensions, vessels with varying cross-section, centrifuges and several space dimensions, as well as its current applications are reviewed.