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Many microorganisms live and evolve in complex fluids. Examples include mammalian spermatozoa in cervical mucus, worms (e.g., \textit{C. elegans}) in wet soil, and bacteria (e.g., \textit{H. pylori}) in our stomach lining. Due to the presence of (bio)polymers and/or solids, such fluids often display nonlinear response to (shear) stresses including viscoelasticity and shear-rate dependent viscosity. The successful interaction between these microorganisms and their fluid environment is critical to the function of many biological processes including human reproduction, ecosystem dynamics, and the spread of disease \& infection. This interaction is often nonlinear and can lead to many unexpected behavior. Here, I will discuss developments in characterizing, modeling, and understanding the swimming behavior of model microorganism in viscoelastic and shear-thinning fluids. Three main microorganisms will be explored: (i) the nematode \textit{C. elegans}, an undulatory swimmer; (ii) the green algae \textit{C. reinhardtii}, a puller swimmer; and (iii) the bacterium \textit{E. coli}, a pusher swimmer. Investigation with artificial particles/swimmers will also be discussed; such studies are helpful in decoupling the biology from hydrodynamic effects. We will explore the interactions between these swimmers' gaits, geometry, and actuation and fluid rheological behavior using mostly experiments, and discuss these results relative to numerical and analytical predictions.