Suspensions of swimming microorganisms are characterized by complex dynamics involving strong fluctuations and large- scale correlated motions. These motions, which result from the many-body interactions between particles, are biologically relevant as they impact mean particle transport, mixing and diffusion, with possible consequences for nutrient uptake. Using direct numerical simulations, I first investigate aspects of the dynamics and microstructure in suspensions of interacting self-propelled rods at low Reynolds number. A detailed model is developed that accounts for hydrodynamic interactions based on slender-body theory. It is first shown that aligned suspensions of swimming particles are unstable as a result of hydrodynamic fluctuations. In spite of this instability, a local nematic order persists in the suspensions over short length scales and has a significant impact on the mean swimming speed. Consequences of the large-scale orientational disorder for particle dispersion are discussed and explained in the context of generalized Taylor dispersion theory. Dynamics in thin liquid films are also presented, and are characterized by a strong particle migration towards the interfaces. The results from direct numerical simulations are then complemented by a kinetic model, in which the dynamics are captured using a conservation equation for the particle configurations, coupled to a mean-field description of the flow arising from the active stress exerted by the particles on the fluid. Based on this model, the stability of isotropic suspensions of particles is investigated. I demonstrate the existence of an instability in which shear stresses are eigenmodes and grow exponentially at long scales, and propose an interpretation in terms of the system entropy. Non-linear effects are also studied using numerical simulations of the kinetic equations in two dimensions. These simulations confirm the results of the stability analysis, and the long-time non-linear behavior is shown to be characterized by the formation of strong density fluctuations, which merge and break up in time in a quasi- periodic fashion. These complex motions result in very efficient fluid mixing, which is quantified by means of a multiscale mixing norm.