In the forty years since the publication of the first edition, this textbook has managed to become a classic. Much attention is paid to the geometric meaning of the basic concepts. The book traces the close relationship of the subject with applications, especially with mechanics. The emphasis in the presentation is not on formulas, but on the geometric meaning of the basic definitions and theorems .. The author introduces the reader to such concepts as manifolds, single-parametric groups of diffeomorphisms, tangential spaces and bundles. Among the examples considered from mechanics are the study of phase portraits of conservative systems with one degree of freedom, the theory of small oscillations, parametric resonance. The book is intended for students and postgraduates of mathematical faculties of universities and universities with an expanded program in mathematics .