Controlled stochastic processes with discrete time form a very interest- ing and meaningful field of research which attracts widespread attention. At the same time these processes are used for solving of many applied problems in the queueing theory, in mathematical ecomics. in the theory of controlled technical systems, etc. . In this connection, methods of the theory of controlled processes constitute the every day instrument of many specialists working in the areas mentioned. The present book is devoted to the rather new area, that is, to the optimal control theory with functional constraints. This theory is close to the theory of multicriteria optimization. The compromise between the mathematical rigor and the big number of meaningful examples makes the book attractive for professional mathematicians and for specialists who ap- ply mathematical methods in different specific problems. Besides. the book contains setting of many new interesting problems for further invf'stigatioll. The book can form the basis of special courses in the theory of controlled stochastic processes for students and post-graduates specializing in the ap- plied mathematics and in the control theory of complex systf'ms. The grounding of graduating students of mathematical department is sufficient for the perfect understanding of all the material. The book con- tains the extensive Appendix where the necessary kwledge ill Borel spaces and in convex analysis is collected. All the meaningful examples can be also understood by readers who are t deeply grounded in mathematics.