Synopsis This book is designed to be a concise and rigorous introduction to the theory of functions of a complex variable, suitable for a one-semester course. The aim of the notes is to help students of mathematics and related sciences acquire a basic understanding of the subject, as a preparation for pursuing it at a higher level or for employing it in other areas. The approach is standard and somewhat old-fashioned. This book should be accessible to students in mathematics and the physical sciences who have previously studied the theory of functions of a real variable. The user of the notes is assumed to have a thorough grounding in basic real analysis. Notions like metric, open set, closed set, interior, boundary, limit point, and uniform convergence are employed without explanation. Especially important are the notions of a connected set and of the connected components of a set. Basic notions from abstract algebra also occur now and then.