The book "Geometric ideas of Riemann and their modern development" by B. F. Kagan is a fascinating journey into the world of higher mathematics, which will not leave indifferent neither students nor teachers, nor all those who are interested in geometry and its applications in various fields of science. In this work, the author masterfully combines the historical context with modern achievements, revealing to the reader the depth and versatility of the ideas inherent in the works of the great mathematician Bernhard Riemann. The book begins with an introduction to the basic concepts of Riemannian geometry, which allows the reader, even who does not have deep knowledge in mathematics, to gradually immerse himself in the topic. Kagan explains how Riemann's ideas on manifolds and metrics became the basis for further research in differential geometry and topology. It not only describes historical facts, but also demonstrates how these ideas continue to evolve and be applied in modern scientific research, such as relativity theory and quantum physics. For whom is this book intended? First of all, it will be of interest to students and graduate students of mathematical and physical faculties, as well as teachers who are looking for inspiration for their lectures . However, even those who are just interested in mathematics or want to expand their horizons will find in it a lot of useful and fascinating. The book is written in accessible language, which makes complex concepts more understandable and less frightening for a wide audience. The topics raised in the book cover not only mathematical aspects, but also philosophical reflections on the nature of space and time. Kagan raises questions about how geometry affects our perception of the world and how it can be used to solve practical problems. It explores the relationship between geometry and other areas of knowledge, such as physics, computer science, and even artificial intelligence, making the book particularly relevant in light of current scientific trends. Kagan's style can be described as clear and logical, allowing the reader to easily follow his thoughts. He skillfully combines strict scientificity with a lively and fascinating presentation, which makes reading not only informative, but also pleasant. In. F. Kagan is the author of several other well-known works, including "Differential Geometry" and "Mathematical Physics", which also deserve the attention of mathematics lovers. Readers interested in works such as "Geometry and Physics" P. Penrose or "