The book is based on the lecture course "Function spaces", which the author gave for more than 10 years in the People's Friendship University of Russia (Moscow). The idea to write this book was proposed by Professors H. Triebel and H.-J. SchmeiBer in May-June 1993, when the author gave a short lecture course for post-graduate students in the Friedrich-Schiller University Jena. The initial plan to write a short book for post-graduate students was trans­ formed to wider aims after the work on the book had started. Finally, the book is intended both for graduate and post-graduate students and for researchers, who are interested in applying the theory of Sobolev spaces. Moreover, the methods used in the book allow us to include, in a natural way, some recent results, which have been published only in journals. Nowadays there exist numerous variants and generalizations of Sobolev spaces and it is clear that this variety is inevitable since different problems in real analysis and partial differential equations give rise to different spaces of Sobolev type. However, it is more or less clear that an attempt to develop a theory, which includes all these spaces, would not be effective. On the other hand, the basic ideas of the investigation of such spaces have very much in common.