Textbook in PDF format This book shows the scope of analytic number theory both in classical and modern directions. There are no division lines; in fact our intentis to demonstrate, particularly for newcomers, the fascinating countless interrelations. Of course, our picture of analytic number theory is by no means complete, but wetried to frame the material into a portrait of a reasonable size, yet providing a self-contained presentation. We were writing this book in a period of time during and after teaching courses and working with graduate students in Rutgers University, Bordeaux University and Courant Institute. We thank these institutions for providing conditions for both of us to work together. We shared ideas on what this book should be about with many of our colleagues, who gave us critical suggestions. Among them we would like to mention Etienne Fouvry, John Friedlander, Philippe Michel and Peter Sarnak. During along process of typing and preparation of this book for publication, We received stimulating encouragement and technical advice from Sergei Gelfand, for all of his help We express our gratitude. Carol Hamer helped to polish some of our English phrases while her little boys tried to destroy the TeX files without success. We thank them all for the output