Were detected have been corrected. This is essentially a textbook for a modern course on differential geometry and topology, which is much wider than the traditional courses on classical differential geometry, and it covers many branches of mathematics a knowledge of which has now become essential for a modern mathematical education. We hope that a reader who has mastered this material will be able to do independent research both in geometry and in other related fields. To gain a deeper understanding of the material of this book, we recommend the reader should solve the questions in A.S. Mishchenko, Yu.P. Solovyev, and A.T. Fomenko, Problems in Differential Geometry and Topology (Mir Publishers, Moscow, 1985) which was specially compiled to accompany this course.