This book is an introduction to the elementary structure theory of finite state machines. After preliminary definitions, the book discusses what it means for one machine to implement another, using the notation of a realization, and in particular, the notion of machine reduction. Reference is made to the fact that machines form a category, with realizations as morphisms. Next, the serial and parallel composition of machines are looked at and it is shown that these operations satisfy certain algebraic relations up to isomorphism. The Hartmanis-Yoeli parallel and serial decomposition theorems, which depend on congruence relations of the state space of a machine are then presented. Subsequent chapters investigate the computation of the set of all such congruences, making use of the fact that they form a lattice. Finally, the lattice of congruences are used to analyze machine decomposition.