This is a text for a first course in complex variable theory designed to follow a course in real variables such as may be found in Advanced Real Calculu8. * It is designed for upper class mathematics majors and beginning graduate students. Notes on which this book is based have also been successfully used for graduate engineering students.The level of rigor is on a par with that found in ARC, that is, an exact E, 8-treatment. Keeping in mind the class of readers to whom the book is addressed, we have made the exposition detailed and uniform throughout the whole text. The only important theorem that we do not prove is the Jordan separation theorem, which, however, is extensively discussed. It is the opinion of the author that a logically satisfying proof of this theorem would take us too far afield. The interested reader can quickly see that for our practical purposes it is not necessary.