This paper presents a new concept of sets which we call it transversal (upper, lower or medial) sets. We introduce this concept as a natural extension of ordinary, fuzzy and transversal fuzzy sets. Transversal sets are a new way in the nonlinear analysis. Applications in minimax theory, algebra, topology, analysis, games theory, algebraic equations theory, ideal theory of BCC-algebras, theory of measures and integration, and convex analysis are considered. First time in history of the theory sets, we give a technology of an arbitrary set, in the sense that every set has ”three sides” which are invisible but they de facto existing.