Swimming microorganisms such as bacteria, spermatozoa, algae, and nematodes are critical to ubiquitous biological phenomena such as disease and infection, ecosystem dynamics, and mammalian fertilization. While there has been much scientific and practical interest in studying these swimmers in Newtonian (water-like) fluids, there are fewer systematic experimental studies on swimming through non-Newtonian (non-water-like) fluids with biologically-relevant mechanical properties. These organisms commonly swim through viscoelastic, structured, or shear-rate-dependent fluids, such as blood, mucus, and living tissues. Furthermore, the small length scales of these organisms dictate that their motion is dominated by viscous forces and inertia is negligible. Using rheology, microscopy, particle tracking, and image processing techniques, we examine the interaction of low Reynolds number swimmers and non-Newtonian fluids including viscoelastic, locally-anisotropic, and shear-thinning fluids. We then apply our understanding of locomotion to the study of the genetic disease Spinal Muscular Atrophy.