The book "To the compactness theorem for differential forms" by the authors Kuzminov V. and . Shvedova I . A. is a fascinating and in-depth study in mathematical analysis and differential geometry. This work will be a real find for students, graduate students and anyone interested in modern mathematical concepts, especially in the context of the theory of differential forms and their application in various fields of science. The book focuses on the compactness theorem, which plays a key role in analysis and geometry. The authors examine in detail its meaning, as well as the consequences arising from this theorem. They not only explain theoretical aspects, but also offer the reader practical examples, which makes the material accessible and understandable. The book is filled with mathematical calculations that help to understand more deeply how the compactness theorem can be applied in a variety of contexts, including physics and engineering sciences. This work will be especially interesting for those who already have basic knowledge in the field of higher mathematics and seek to expand their horizons. Students of mathematical and physical faculties, graduate students, as well as teachers will find in it many useful ideas and approaches. The book is suitable for both self-study and for use in the educational process, as it contains many tasks and exercises that will help to consolidate the knowledge gained. The topics raised in the book cover not only theoretical aspects, but also practical applications of differential forms. The authors investigate how compactness affects the properties of functions and shapes, which is important for understanding more complex mathematical structures. The book also deals with issues related to topology and analysis, which makes it relevant for a wide range of readers interested in modern mathematical directions. The style of Kuzminov and Shvedov is clear and logical They skillfully combine theoretical calculations with practical examples, which allows the reader not only to assimilate the material, but also to see its application in real life. Both authors have significant experience in teaching and research, which makes their work particularly valuable for students and researchers. Kuzminov V. and . is known for his works in the field of functional analysis, and Swedov I. A. his research in the field of differential geometry and mathematical physics . The book "To the compactness theorem for differential forms" is not just a textbook, but also a real guide to the world of modern mathematical research. It encourages the reader to reflect on the deep connections between the various fields of mathematics and their practical applications. Readers interested in works such as "