Summary: This paper provides a framework for deriving necessary conditions, in the form of a maximum principle, for minimax optimal control problems. The distinguishing feature of these problems is that the data depends on a vector \(\alpha\) of unknown parameters, and ``optimality'' is defined on a worst case basis, as \(\alpha\) ranges over the parameter set \({\mathcal A}\). The centerpiece, a minimax maximum principle, is a set of optimality conditions for such problems. Here, the parameter set \({\mathcal A}\) is taken to be an arbitrary compact metric space and the hypotheses imposed on the dynamics and endpoint constraints are of an unrestrictive nature. The minimax maximum principle captures as special cases necessary conditions for optimal control problems with minimax costs, for problems involving ``semi-infinite'' endpoint constraints, and also a maximum principle for state constrained optimal control problems.