This book is intended as a first course in the theory of functions of a single complex variable for students (of Engineering, Physics or Mathematics) who have already acquired the fundamental notions of real Mathematical Analysis. It is the English version of an older book, published by Springer in Italian, based on the lectures given by the author to the students of the University of Rome La Sapienza and of the Scuola Superiore di Catania. The claim of mathematical rigor is accompanied by the idea that students should also develop know-how skills: half of the book is devoted to the detailed solution of 288 exercises illustrating, also with figures, the theory. Also, the choice of the topics reflects the above point of view.