During the last twenty years, the interplay between group theory, geometry and formal language theory has led to far-reaching developments in group theory. In particular, Gromov's theory of hyperbolic groups has played a prominent role. Recently, Gilman has given a language-theoretic characterisation of these groups, and he and Duncan have adapted this approach to initiate a theory of `hyperbolic' semigroups. The talk will give an introduction to these ideas, and report on joint work with Mark Kambites on hyperbolic completely semigroups.