This book explores, from an elementary point of view, a number of so-called variational problems. In these problems, the values dependent on the curve are considered, and the curve for which this value reaches its highest or lowest value. is sought. These are, for example, the problems: among all the curves connecting two points on a surface, find the shortest; on the plane among all closed curves of a given length, find the one that limits the greatest area, and t. d. The material of this book was mainly presented by the author at lectures in the school mathematical circle of Moscow State University .. The content of the first lecture (1-10) basically coincides with the content of the published in 1940 .. Author's brochures "Geodesic Lines" . The reader is expected only to get acquainted with the course of elementary mathematics .. At the same time, the first chapters are absolutely elementary in nature, while others, without requiring special knowledge, require a little more skill in mathematical reading and thinking. The whole material of the book can be considered as an elementary introduction to variational calculus (this is the name of the branch of mathematics in which problems are systematically studied to find a minimum or maximum of functionals). Variational calculus is not included in the first concentration of the course of "higher mathematics", studying, for example, in technical universities. However, we believe that for a person starting a course in "higher mathematics", it is not useless to look further ahead. For the reader familiar with the elements of mathematical analysis, it will not be difficult to make some definitions and reasonings set forth in the book are not strictly, absolutely strict (explaining considerations for this he will often find in the text given in small print). It is necessary, for example, to speak not of small quantities and their approximate equality, but of infinitely small quantities and their equivalence. If a more exacting reader remains unsatisfied with the level of rigor and logical completeness of the reviews allowed here , Let this serve for him as an explanation of the need for that logical grinding of the basic concepts of mathematical analysis. which he will face.... for example, the University Courses in Analysis . Without this grinding, a strict and systematic presentation of such chapters of analysis as variational calculus is impossible. Mathematical analysis has developed a powerful analytical apparatus, sometimes automatically solving many difficult problems. However, at all stages of mastering mathematics, it is extremely important to see the simple geometric or physical meaning of the problem being solved. You need to be able to solve problems "on your fingers", as mathematicians say, t. e. Find an easy but easy way to prove it................. If this small book will at least contribute to the development of these elements of mathematical culture among readers, then the author will consider that the work he spent on writing it was not useless.