The book is devoted to the most important problems of optimization. It is built on the basis of teaching the theory of optamization at the Faculty of Mechanics and Mathematics of Moscow State University. It is based on courses and special courses taught by E. M. Galeyev . Fragments of the following sections of the theory of extreme problems of linear and convex programming, mathematical programming, classical variational calculus and optimal control are considered: Both necessary and sufficient extremum conditions are given. To study these sections to the required extent, elements of functional and convex analysis are given. In each paragraph, after the theoretical part, examples of problem solving are given, problems are proposed for solving in seminars, control and for homework. An overview of the general methods of extremum theory. For university students in the specialties "Mathematics", "Applied Mathematics", as well as for graduate students, teachers and researchers .