  This paper has several purposes. We present through a critical review the
results from already published papers on the constructive semigroup theory, and
contribute to its further development by giving solutions to open problems. We
also draw attention to its possible applications in other (constructive)
mathematics disciplines, in computer science, social sciences, economics, etc.
Another important goal of this paper is to provide a clear, understandable
picture of constructive semigroups with apartness in Bishop's style both to
(classical) algebraists and the ones who apply algebraic knowledge.
