This book presents a series of challenging mathematical problems which arise in the modeling of NonNewtonian fluid dynamics. It focuses in particular on the mathematical and physical modeling of a variety of contemporary problems and provides some results.ampampnbspThe flow properties of NonNewtonian fluids differ in many ways from those of Newtonian fluids. Many biological fluids blood for instance exhibit a nonNewtonian behavior as do many naturally occurring or technologically relevant fluids such as molten polymers oil mud lava salt solutions paint and so on.ampampnbspThe term complex flows usually refers to those fluids presenting an internal structure fluid mixtures solutions multiphase flows and so on.