The book is an introduction to the theory of entire and meromorphic functions intended for advanced graduate students in mathematics and for professional mathematicians. The book provides a clear treatment of the Nevanlinna theory of value distribution of meromorphic functions, starting from scratch. It contains the first book-form presentation of the Rubel-Taylor Fourier series method for meromorphic functions and the Miles theorem on efficient quotient representation. It has a concise but complete treatment of the Polya theory of the Borel transform and the conjugate indicator diagram. It contains some of Buck's results on integer-valued entire functions, and the Malliavin-Rubel uniqueness theorem. The book closes with applications to mathematical logic. In particular, the first-order theory of the ring of entire functions is developed and questions concerning identities of exponential functions are studied as in Tarski's "High School Algebra Problems." The approach of the book gets to the heart of the matter without excessive scholarly detours. It prepares the reader for further study of the vast literature on the subject, which is one of the cornerstones of complex analysis.