This course discusses the formulation and the solution techniques to a wide ranging class of optimal control problems through several illustrative examples from economics and engineering, including: Linear Quadratic Regulator, Kalman Filter, Merton Utility Maximization Problem, Optimal Dividend Payments, Contact Theory. The method of dynamic programming and Pontryagin maximum principle are outlined. Brief introductions to the general theory of backward and forward stochastic differential equations, related partial differential and viscosity theory are discussed again through the above examples.