The present lecture notes survey the combined inertia-free motion of mesoscale particles and dispersing fluid under low Reynolds number conditions. The survey starts with a discussion of relevant time scales, progressing then to methods to obtain flow solutions of the fundamental Stokes equation for isolated rigid particles such as spheres and rods. Kinematic reversibility is discussed, as well as the peculiar hydrodynamic propulsion of active microswimmers by non-reciprocal, periodical changes of their body shape. The three-beads model and a simple sperm model are analyzed as examples, respectively, of an artificial and biological microswimmer. The fluid-mediated hydrodynamic interactions (HIs) between moving particles are thoroughly discussed, in particular regarding the settling of two-, three- and many-particle clusters under gravity. HIs between three and more settling particles give rise to horizontal mixing, the appearance of chaotic particle trajectories, and a mean settling velocity of the cluster, which for an infinite fluid increases with increasing number of cluster particles. It is shown how the mobility matrices characterizing the HIs for a given particle configuration can be calculated for spherical particles. These matrices are then used for the calculation of the effective viscosity, and the mean sedimentation velocity in homogeneous suspensions of colloidal particles. The interplay of HIs and Brownian motion leading to non-Newtonian flow behavior of colloidal hard-sphere suspensions is analyzed. The hindered particle sedimentation in a homogeneous suspension is shown to arise from the effect of fluid backflow, in combination with hydrodynamic and direct particle interactions. While the difference between the steady shear viscosity and its (short-time) high frequency part becomes large for concentrated suspensions, the difference between long-time and short-time sedimentation coefficients remains very small.