The book was written by a major mathematician Richard Courant in collaboration with Herbert Robbins. It is intended to close the gap between mathematics, which is taught at school, and the most lively and important sections of natural science and technology in modern mathematical science. Starting with elementary concepts, the reader moves to important areas of modern science. The book is written in an accessible language and is a classic of a popular genre in mathematics. The book is intended for schoolchildren, students, teachers, as well as for everyone interested in the development of mathematics and its structure. In this edition, some errors have been fixed and previously omitted text fragments have been restored.