This chapter focuses on a set of optimality conditions known as the Maximum Principle. Many competing sets of optimality conditions are now available, but the Maximum Principle retains a special significance. An early version of the Maximum Principle due to Pontryagin er al. was after all the breakthrough marking the emergence of Optimal control as a distinct field of research. Also, whatever additional information about minimizers is provided by Dynamic Programming, higher-order conditions, and the analysis of the geometry of state trajectories, first-order necessary conditions akin to the Maximum Principle remain the principal vehicles for the solution of specific optimal control problems (either directly or indirectly via the computational procedures they inspire), or at least for generating “suspects” for their solution.