For the first time in the book, a systematic presentation of the properties of the Rademacher system is given primarily from the point of view of function theory and functional analysis. The structure of the book is subordinate to its main idea - to show the relationship of the properties of the Rademacher system with the geometry of the functional spaces containing it; it consists of three parts, in which the Rademacher system is studied respectively in Lp-spaces, in general symmetric spaces and in some classes of asymmetric spaces (BMO, Paley, Cesaro, Morry). The material of several chapters of the book served as the basis for special courses read to students of Samara State University for several years. The book is intended for researchers, as well as senior students and graduate students studying in relevant specialties.