This book deals with PDE models for chemotaxis (the movement of biological cells or organisms in response of chemical gradients) and hydrodynamics (viscous, homogeneous, and incompressible fluid filling the entire space). The underlying Keller-Segel equations (chemotaxis), Navier-Stokes equations (hydrodynamics), and their numerous modifications and combinations are treated in the context of inhomogeneous spaces of Besov-Sobolev type paying special attention to mapping properties of related nonlinearities. Further models are considered, including (deterministic) Fokker-Planck equations and chemotaxis Navier-Stokes equations. These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov-Sobolev type and interested in mathematical biology and physics.