The book is dedicated to the most important optimization problems. It is built on the basis of teaching the theory of optimization at the Faculty of Mechanics and Mathematics of Moscow State University. It is based on courses taught in 199899 by E.M. Galeev (Chapters 1-5) and V.M. Tikhomirov (Chapter 6). Fragments of the following sections of the theory of extreme problems are considered: linear and convex programming, mathematical programming, classical calculus of variations and optimal control.