Recently large collections of swimming microorganisms have been observed producing collective motions on a scale much larger than the scale of a single organism. To better understand the cause of these motions, simulations of large populations of hydrodynamically interacting swimming particles have been performed at low Reynolds number in periodic and confined geometries. Each swimmer is modeled as a rod containing beads with a propulsion force exerted on one bead (with an equal and opposite force exerted on the fluid) and excluded volume potentials at the beads. At small concentrations, the swimmers behave analogously to a dilute gas in which the hydrodynamic interactions perturb the ballistic trajectories into diffusive motion. Simple scaling arguments can explain the swimmer behavior as well as the behavior of passive tracer particles. As the concentration increases, the multi-body hydrodynamic interactions lead to large-scale collective motion. This collective motion manifests itself as a diverging correlation length and diverging velocity fluctuations with system size.