In this chapter elementary examples illustrate the basic concepts of the theory of difference schemes: approximation, stability and convergence. The reader is already familiar with several methods available for investigating stability and convergence such as the method of separation of variables and the method of energy inequalities. In Section 4 difference equations are treated as operator equations in an abstract space. These provide a means of studying a wide range of interesting methods in a unified manner. We begin our exposition with a discussion of examples that make it possible to draw fairly accurate outlines of the possible theory regarding these questions and with a listing of the basic results of the book together with a development desired for them.