This book deals with homogeneous function spaces of Besov–Sobolev type within the framework of tempered distributions in Euclidean n-space based on Gauss–Weierstrass semi-groups. Related Fourier-analytical descriptions and characterizations in terms of derivatives and differences are incorporated afterwards as so-called domestic norms. This approach avoids the usual ambiguities modulo polynomials when homogeneous function spaces are considered in the context of homogeneous tempered distributions. 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. In particular it might be of interest for researchers dealing with (nonlinear) heat and Navier-Stokes equations in homogeneous function spaces.