We suppose that X and Y are nonempty sets and f : X x Y --> R. A minimax theorem is a theorem which asserts that, under certain conditions,

The original motivation for the study of minimax theorems was, of course, Von Neumann's 1928 work on games of strategy. After a lapse of nearly ten years, generalizations of Von Neumann's original result for matrices started appearing. As time went on, these generalizations became progressively more remote from game theory, and minimax theorems started becoming objects of study in their own right.

In this article, we trace the development of minimax theorems starting from Von Neumann's original result. We discuss infinite dimensional bilinear results and their connection with weak compactness. We discuss the results for concave-convex functions, and their generalizations to quasiconcave-quasiconvex functions.

We discuss various minimax theorems in which X and Y are not assumed to be subsets of vector space. These fall naturally into three classes, topological minimax theorems in which various connectedness hypotheses are assumed for X, Y and f, quantitative minimax theorems in which no special properties are assumed for X and Y, but various quantitative properties are assumed for f and, finally, mixed minimax theorems in which the quantitative and the topological properties are mixed.

The recent unifying metaminimax theorems, theorems which imply simultaneously the minimax theorems of all the above three types and which depend on abstract generalizations of connectedness tend to show that the classification above is perhaps too rigid.

We also discuss minimax inequalities for two or more functions and the connections between minimax theorems and certain fixed-point and coincidence theorems.