Until this juncture, our development of the necessary and sufficient conditions of optimal control theory was essentially a reformulation of those from the classical calculus of variations. As a result, the power and reach of optimal control theory have not been fully exposed or exploited. The goal of this chapter, therefore, is to state and prove some necessary and sufficient conditions for a class of control problems that permit the full capability of optimal control theory to be realized. The theorems are not the most general we will encounter, but they do highlight the motivation for the name Maximum Principle. Moreover, our proof of the necessary conditions will employ some continuity assumptions that are, strictly speaking, not needed for a rigorous proof, but will nonetheless be employed so as to ease the technical burden and bring in some connections with the principle of optimality and dynamic programming. We will then use the necessary conditions to explicitly solve for the optimal paths of some examples, some of which will be devoid of any economic content so as to emphasize how to arrive at a solution in practice. The reader is encouraged to work through the proof of the necessary and sufficient conditions, as it provides the reader with a better understanding of how they differ from those presented earlier. Note that the theorems are stated and proven for a class of control problems with many state and control variables.