The mathematical formulation of the most general problem of Optimal Control can be considered as a problem of Mayer subjected to unilateral constraints, i.e., to certain restrictions expressible in terms of inequalities. Results of the classical calculus of variations in their usual forms cannot give a general solution to this problem because, among other things, the fundamental relation of the calculus of variations, i.e., the equation of Euler- Lagrange, is valid only in the case of points interior to the set of admissible points. The most general solution is given by the Maximum Principle of Pontryagin, but in its present form this principle cannot be applied in certain situations, and its validity has been proved in particular cases only. A derivation of this principle for the most general case is given.