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Bombieri s average result on primes in arithmetic progressions is the important theorem which substitutes, in a certain way, the unproved generalized Riemann hypothesis. This has inspired many researchers to investigate more on this kind of results. Our purpose is to gradually introduce the readers to this active area of research. The basic definitions and notations are thoroughly given. All basic results used in this book are listed with references. A short history and the developement of this subject are discussed. Two new results on the second moment associated with the divisor function in arithmetic progressions are obtained. The large sieve along with two exciting methods of Hooley and Vaughan are our main tools and will be illustrated. Assuming only the material in an introductory course of analytic number theory, this book is therefore suitable for both graduate students and researchers working in the area of number theory.