The book “Graded Algebras and Hilbert’s 14th Problem” by Ivan Vladimirovich Arzhantsev is a unique textbook that opens the door to the world of abstract algebra and the theory behind one of the most intriguing problems of mathematics of the 20th century. This work will be a real find for students, graduate students and anyone who is interested in modern mathematical concepts, and also wants to better understand the structure and properties of graduated algebras. In his manual, Arzhantsev not only outlines the theoretical foundations of graded algebras, but also connects them with Hilbert's 14th problem, which concerns the existence and properties of algebraic structures with certain properties. This problem, formulated at the beginning of the 20th century, still remains relevant and causes interest among mathematicians around the world. The author masterfully guides the reader through the labyrinths of complex concepts, emphasizing their practical application and significance in various areas of mathematics. The book will be especially interesting for those who are passionate about mathematics and want to expand their knowledge in the field of algebra and category theory. Graduate students and graduate students studying algebra, representation theory and related disciplines will find useful materials in this manual that will help them to understand the subject more deeply. The book can also attract the attention of teachers who want to enrich their courses with modern approaches and ideas. The topics raised in the book cover a wide range of issues, from the basics of graded algebras to their application in various mathematical theories. Arzhantsev focuses on the importance of understanding the structure of these algebras for solving more complex problems, which makes his textbook not just a teaching material, but a real guide to the world of high mathematical ideas. The book also includes many examples and tasks, which allows the reader not only to learn the theory, but also to apply knowledge in practice. The style of Ivan Vladimirovich Arzhantsev is clear and accessible, which makes even the most complex concepts understandable to a wide audience. It skillfully combines theoretical calculations with practical examples, which allows the reader not only to assimilate the material, but also to see its application in real problems. Arzhantsev is not only the author of this book, but also a recognized expert in the field of algebra, and his works, such as “Algebraic structures and their application”, have already won recognition among specialists. If you are looking for literature that will help you understand complex mathematical concepts and give you the opportunity to look at algebra from a new perspective, “Graded Algebras and Hilbert’s 14th Problem” is what you need. This book will not only be a useful tutorial, but also a source of inspiration for those who seek a deep understanding of mathematics and its fundamental issues. Thus, if you are a student, a graduate student or just a math enthusiast who wants to expand your horizons and deepen your knowledge in the field of algebra, do not miss the opportunity to get acquainted with this work. It will not only enrich your wealth of knowledge, but also help you better understand the complex relationships in mathematics that continue to excite the minds of scientists around the world.