Microorganisms such as bacteria and spermatozoa move in a world where viscous forces completely dominate inertial forces, and the time evolution of their motion may be thought of as a sequence of steady state snapshots. In this world, what motility strategies give rise to efficient locomotion? The study of the fluid dynamics of microorganism motility began with the classic work of G.I. Taylor in 1951, and has been an active area of research in the last decades. Current modeling challenges include the collective dynamics of microorganisms and their interactions with surrounding physical and chemical environments, coupling of their internal force-generating mechanisms with external fluid dynamics, as well as their motion through viscoelastic fluids. We will present recent work that sheds light on these complex systems.