Countably compact topological semigroups

We shall give a survey of the development of the theory of compact and compact-like topological semigroups from Wallace's Problem up to the modern time. We shall discuss on the following directions:
(i) the closure of subgroups in countably compact and pseudocompact topological semigroups;
(ii) the continuity of the inversion in countably compact and pseudocompact topological semigroups; and
(iii) embeddings of topological semigroups into countably compact (pseudocompact) topological semigroups.