As one can see from the title, this book is an introduction to the study of diophantine equations. The material is organized in two parts. The first part contains three chapters. Chapter 1 introduces the reader to the main elementary methods in solving diophantine equations such as decomposition, modular arithmetic, mathematical induction, Fermat's infinite descent. Chapter 2 presents some classical diophantine equations, including linear, pythagorean and some higher degree equations. Chapter 3 focuses on Pell's-type equations, serving again as an introduction to this special class of quadratic diophantine equations. Throughout Part I, each of the sections contains representative examples that illustrate the theoretical part.