This article is a survey of recent results on non-linear minimax problems. The following questions are considered: The directional differentiability of the maximum function;necessary conditions for a minimax and their geometrical interpretation; sufficient conditions for a local minimax; methods of successive approximation to find the stationary points of the maximum function; properties of the maximin function. These questions are set out first of all for the discrete case (Ch. I) and then for the general case (Ch. II); in the first chapter the accent is on methods of successive approximation, while in the second it is on the tie-up between the theory, as it has evolved, and certain classical results.