This book deals with the most important problems of optimization. It is based on courses and special courses on optimization theory, read by the author on. Faculty of Mechanics and Mathematics of Moscow State University . Fragments of the following sections of the theory of extreme problems are considered: linear and convex programming, mathematical programming, classical variational calculus and optimal control. 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 at seminars, in control works, as well as for self-acquisition of material . An overview of the general methods of extremum theory. For university students studying in the specialties "Mathematics", "Applied Mathematics", as well as for graduate students, teachers and researchers .