In this talk, we use biological locomotion on small scales as an inspiration (and an excuse) to solve a number of modeling problems in small-scale fluid mechanics. We first solve for the swimming kinematics of elastic swimmers, devices which exploit flow-induced deformation of elastic filaments for propulsion. More generally, we then show how soft surfaces can be exploited for propulsion without inertia. Finally, we describe how the viscoelastic nature of the surrounding fluid can affect the kinematics and energetics of simple swimmers.