This course is built in accordance with the programs of mechanical-mathematical and physical-mathematical faculties of universities and pedagogical institutes. The difference between these programs is reflected in the fact that a number of paragraphs, paragraphs and one chapter of the book are marked with an asterisk. When using the course in pedagogical institutes, all the material marked in this way can be released, which does not affect the integrity of the rest of the presentation. The book differs from its first edition (Differential Geometry, Uchpedgiz, 1948) by some rearrangement of the material, minor additions, a change in some designations and a change in the principles of numbering paragraphs and formulas.