(Lobachevskii Journal of Mathematics, Volume XVIII)
In this paper we introduce the notion of a universal semigroup and its dual, the universal cosemigroup. We show that the class of universal semigroups include the class of monoids and is included in the class of semigroups with a product that is an epimorphism. Both inclusions are proper. Semigroups in the category of Banach spaces are Banach algebras and we show that all Banach algebras with an approximate unit are universal and construct a finite dimensional Banach algebra that has no unit but is universal. The property of being universal is thus a generalized unit property.