This book covers the basic elements of difference equations and the tools of difference and the sum calculus necessary for studying and solving, primarily, ordinary linear difference equations. It is lucidly written and carefully motivated with examples from various fields of applications. These examples are presented in the first chapter and then discussed with their detailed solutions in Chapter 2. A particular feature is the use of the discrete Fourier transforms for solving difference equations associated with, generally nonhomogeneous, boundary conditions. Emphasis is placed on illustrating this new method by means of applications. The primary goal of the book is to serve as a primer for a first course in linear difference equations but, with the addition of more theory and applications, the book is suitable for more advanced courses.