Automata and semigroups recognizing infinite words

This paper is a survey on the algebraic approach to the theory of automata accepting infinite words. We discuss the various acceptance modes (Büchi automata, Muller automata, transition automata, weak recognition by a finite semigroup, omega-semigroups) and prove their equivalence. We also give two algebraic proofs of McNaughton's theorem on the equivalence between Büchi and Muller automata. Finally, we present some recent work on prophetic automata and discuss its extension to transfinite words