Since 1970 some late but important progress has been made in Fourier Analysis of singular measures. Many results and methods should become classical. This volume was written as an introduction to the subject, though still retaining the form of a research paper, emphasizing the point of view and the work of the authors. The theory rest on some simple general properties of $L^{\infty}$-spaces and semi-topological semi-groups, presented in the preliminaries. The study of measures on locally compact abelian groups is approached in an entirely new context, nevertheless using all the ressources and technical constructions of classical Fourier Analysis, which leads to very precise results. An important part is devoted to the properties at infinity of Fourier-Stieltjes transforms, using a semi-group technique which turns out to be simple and powerful. Another part is devoted to the Gelfand theory of convolution algebras of measures on abelian groups, considerably developping the technique of Sreider's generalized characters. An important place is given to concrete examples, in particu-lar to the detailed study of a class of remarkable measures : the so-called "Riesz products", which provide powerful tools and play a central role in the theory.