A fascinating branch of mathematics since antiquity, the geometry of curves has been extensively developed and become highly abstract. Recently links have been made with the subject of error correction, leading to the creation of geometric Goppa codes, a new and important area of coding theory. This book is an expanded and updated version of one part of the author's successful book Error-Correcting Codes and Finite Fields. Here he gives an elementary introduction to Goppa codes and includes many examples, calculations, and applications. The first part of the book emphasizes motivations, giving precedence to applications over proofs. The second part then provides the formal theory, with some concepts and proofs simplified without sacrificing rigor.