This chapter introduces difference equations and examines some simple but important cases of their applications. It aims to develop simple algorithms for their numerical solutions and apply these techniques to the solution of some problems of interest to the engineering professional. The chapter presents each type of difference equation that is of widespread interest. It explores the general numerical techniques for solving difference equations. The chapter considers some special techniques for obtaining the analytical solutions for the class of linear constant coefficients difference equations. The general solution of a linear difference equation is the sum of its homogeneous solution and its particular solution, with the constants adjusted, so as to satisfy the initial conditions. The direct difference equation formulation is the most amenable to numerical computations because of lower computer memory requirements, while the convolution-summation technique has the advantage of being suitable for developing mathematical proofs and finding general features for the difference equation.