The finite Finite Automata (FA)automata and regular languages have been used in a wide variety of problems in computing, communication, and control, including formal modeling and verification. Finite automata (FA) are recognizers of the simplest kind of the languages—the regular languages. The sentences of these languages are tokens, which have applications in compilers and natural language processing; many modern language processing have also tokenization as a built-in feature. This chapter presents the theory of FA with emphasis on problem solution. The regular expressions and operation on them are introduced in a self-taught style. The design aspects of FA for some applications are also introduced. Other topics, like compilation of regular expressions to build lexical analyzers, countable sets, decidable properties of FA, and recursive and recursively enumerable languages, have been introduced. The number of worked examples have also been added, along with self-review questions and exercises for practice followed by relevant references at the end.