The book is composed of two small and well-compliant works of famous American scientists, It can serve as an initial introduction to the new mathematical discipline, interest in which has grown very much in recent years. The ideas of differential topology have proved extremely fruitful in geometry, in analysis, in the theory of differential equations, as well as in various applications of mathematics. The authors describe the initial concepts of this discipline, illustrating them with a large number of examples. The book should be recommended to anyone starting to study modern mathematics. It is available for undergraduate students of universities and pedagogical institutes, which will also be of interest to both specialists and anyone who wants to get an idea of the mathematics of our days,