We begin this chapter with some notations which are used throughout this monograph. This is followed by some classifications namely: linear and nonlinear higher order d.ifference equations, linear and nonlinear first order difference systems, and initial and boundary value problems. We also include several examples of initial and boundary value problems, as well as real world phenomena from diverse fields which are sufficient to convey the importance of the serious qualitative as well as quantitative study of difference equations. The discrete Rolle's theorem, the discrete Mean value theorem, the discrete Taylor's formula, the discrete !'Hospital's rule, the discrete Kneser's theorem are stated and proved by using some simple inequalities.