Over the last ten years, research in the field of dynamic programming has assumed many different forms. Sometimes, the emphasis has been upon questions of formulation in analytic terms and concepts, sometimes upon the problems of existence and uniqueness of solutions of the functional equations derived from the underlying processes, occasionally upon the actual analytic structure of the solutions of these equations, sometimes upon the computational aspects; and sometimes upon the applications-to control processes, to trajectories of various types, to operations research, to mathematical economics. Inevitably, the result of this quasi-ergodic behavior has been to ignore a number of significant problems, and to treat a number of others in cavalier fashion. In this exposition, we wish to focus attention upon a number of interesting, difficult, and significant questions in analysis which arise naturally out of the functional equation technique of dynamic programming. Our aim is to show that this theory constitutes a natural extension of classical investigations and that the corresponding problems are natural generalizations of problems of classical analysis.