CUR[~b~ FESULTS AND ISSUES IN STOCHASTIC CONTROL A. BENSOUSSAN University Paris Dauphine and INRIA GENERAL INTRODUCTION. Since the basic review of W. FLEMING [I], several surveys and books have ap- peared during the fifteen past years, relating substantial progress of the theory of stochastic control. Among the books, let usmentionW. FLEMING - R. RISHEL [I], A. FRIEDMAN [13, A.N. SHYRYA~ [I] , D. BERTS~KAS - S.E. SHREVE [I], I.I. GI~V~d~N- A.V. SKOROEHOD [I], N.V. KRYLOV [1], M.J. KUSHNER [I], E.B. DYNKIN - A.A. YUSHKEVICH [1], A. BENSOUSSAN [I], A. BENSOUSSAN - J.L. LIONS [1], [2], G. KALLIANPUR [I]. In addition several monographs and lecture notes provide an additional impor- tant material. Among many, let us mention J.M. BISMUT [I], N. ELKAROUI [I], M. NISIO [1], M. HAZEWINKEL - J.C. WILL~S [I], S.K. MII~fER - A. MORO [I], H. KOREZLIOGLU - G. MAZZIO!~fO - J. SZPIRGLAS [1], M.A.H. DI~IPSTER [I]. The Bad Honnef conference proceedings since 1979 represent a good source for the evolution of the art. Let us also mention the three volumes published by the french CNRS, under the responsability of I. LANDAU [I]. Finally the survey of M.H.A DAVIS [I] has been an important guide for the present paper. This abundant litterature showed the importance of the development of the field. The mathematical techniques which are used are quite diversified (partial differen- tial equations, stochastic processes, control theory, algebraic methods, m~aerical analysis...) and the applications are numerous (Engineering, Economics, Operations Research,...). We emphasize in this review the control of diffusions with full or partial ob- servation, for one decision maker and give some hints to other problems. Needless to say, this part of the theory, although of basic importance, does not cover every- thing. Even for it, we do not pretend here to be complete, which would have been im-