The book contains a careful exposition of basic facts concerning manifolds, differential forms and their integration, Stokes theorem, exterior differential systems, Lie derivatives and basic facts on Lie groups and Lie algebras, Riemannian geometry and their use in physics. Comparing the new English version with the original French edition, the last three chapters were substantially reworked and extended (these are chapters on Riemannian geometry, Lagrange and Hamilton mechanics, and symplectic geometry). The book is written in a very understandable and systematic way, with a lot of figures. A very good feature of the book is a collection of more than 130 exercises and problems, which are completely solved, resp. answered. The book can be recommended for a wide range of students as a first book to read on the subject. It can be also useful for the preparation of courses on the topic.