The branch of pure mathematics that primarily deals with the study of the integers and integer-valued functions is referred to as number theory. It includes the study of prime numbers as well as the properties of objects made out of integers. There can be various sub-divisions of number theory. The main sub-divisions of number theory are analytical number theory, algebraic number theory and Diophantine geometry. Some examples of the problems in analytical number theory are the prime number theorem, the Warring problem, Goldbach conjecture, and the Riemann hypothesis. Some of the other subfields of number theory are probabilistic number theory, arithmetic combinatorics and computational number theory. This textbook contains some path-breaking studies on number theory. It traces the progress of this field and highlights some of its key concepts and applications. This book will serve as a reference to a broad spectrum of readers.