This article will appear in the proceedings of the "Clifford Conference", organized in the memory of Professor A.H. Clifford in Tulane (New Orleans) in april 1994. This is a survey article reviewing some recent developments of the theory of semigroups that share the property to have required auxiliary mathematical tools: topology, partial orders and categories. The topics covered in this paper include Reitermans theorem on the definition by identities of varieties of finite semigroups, the extension of Eilenbergs variety theorem to ordered semigroups and the theorems on semigroup expansions for which categories, considered as algebraic objects form the appropriate framework.