Very general conditions are established that ensure the existence of a saddle-value for a function $F(x, y) : C \times D \to \mathbb R$, where $C, D$ are subsets of two topological spaces $X, Y$ , respectively. These conditions are much weaker than those generally required in the literature. As consequences, several minimax theorems are obtained that include as special cases refinements of various minimax theorems developed recently in nonlinear analysis and optimization for quasiconvex quasiconcave functions. Despite the generality of the results, the proof is very simple and is independent of separation or fixed point arguments on which most best known minimax theorems are based.