The book is devoted to the most important problems of extremum theory - mathematical programming, variational calculus and optimal control. The main attention is paid to the Lagrange principle for the necessary conditions, as well as sufficient conditions, convex problems, Hamiltonian formalism. Many problems that have been posed and investigated throughout the history of extremum theory are discussed. The material of the book is based on the experience of teaching the theory of extreme tasks at the Faculty of Mechanics and Mathematics of Moscow State University, and it can serve as a textbook for many different optimization courses. For university students, graduate students, faculty and researchers in mathematics and its applications. Recommended by the Educational and Methodological Council for Mathematics and Mechanics of UMO on classical university education as a textbook for students of higher educational institutions studying in a group of mathematical directions and specialties .