This book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of shastic control. It presents the results in the context of finite and infinite horizon problems, and discusses a number of new and interesting issues. Further, it precisely identifies, for the first time, the interconnections between three well-known, relevant issues – the existence of optimal controls, solvability of the optimality system, and solvability of the associated Riccati equation. Although the content is largely self-contained, readers should have a basic grasp of linear algebra, functional analysis and shastic ordinary differential equations. The book is mainly intended for senior undergraduate and graduate students majoring in applied mathematics who are interested in shastic control theory. However, it will also appeal to researchers in other related areas, such as engineering, management, finance/economics and the social sciences.