Mathematical logic and automata theory are two scientific disciplines with a close relationship that is not only fundamental for many theoretical results but also forms the basis of a coherent methodology for the verification and synthesis of computing systems. Although both automata theory and mathematical logic look back to a much longer history, they have come together in the 1960s through the fundamental work of B¨uchi, Elgot, Rabin and others who showed the expressive equivalence of automata with logical systems such as monadic second-order logic on finite and infinite words and trees. This allowed the handling of specifications (where global system properties are stated) and implementations (which involve the definition of the local steps in order to satisfy the global goals laid out in the specification) in a single framework. Moreover this framework offered algorithmic procedures for essential questions such as the consistency of the specifications or the correctness of implementations. Through the methodology of model-checking the connection between automata theory and logic has indeed become the basis of efficient verification methods with industrial scale applications.