The last twenty years have brought forth a body of research known under the names of the theory of automata and the theory of formal languages. Both topics are closely related and form a distinct part of what is rather nebulously described as "computer science." They are made up of a body of mathematical objects, theorems, and methods of proof that are essentially algebraic, bordering on logic. Many of the published results show motivations specific to the interests of their authors and these were frequently very varied—to mention a few: logic, linguistics, theory of communication, programming, electronic circuits, and even models for neurophysiology. It appeared to me that the time is ripe to try and give the subject a coherent mathematical presentation that will bring out its intrinsic aesthetic qualities and bring to the surface many deep results which merit becoming part of mathematics, regardless of any external motivation. This work, planned in four volumes, of which this is the first, proposes to cover a major part of the field. A serious effort has been made to include most of the interesting results. However, restraint had to be used since many of the ramifications, if pursued too far, would lead to many highly specialized researches, far away from what can be discerned as the principal objectives.