This book is a self-contained account of the theory of viscosity solutions for first-order partial differential equations of Hamilton-Jacobi type and its interplay with Bellman's dynamic programming approach to optimal control and differential games. This is an affordable new softcover edition of a bestselling text replete with exercises. It will be of interest to scientists involved in the theory of optimal control of deterministic linear and nonlinear systems. The work may be used by graduate students and researchers in control theory both as an introductory textbook and as an up-to-date reference book.