We show that how the fundamental theorems on equilibrium problems can be extended to generalized convex spaces. Precisely, most of important results in the KKM theory hold without assuming the linearity in topological vector spaces. Such examples are the KKM theorem, the minimax theorem and the intersection lemma of von Neumann, the Nash equilibrium theorem, various fixed point theorems, Ky Fan’s minimax inequality, variational inequalities, best approximation theorems, existence theorems for solutions of generalized quasi-equilibrium problems, and others.