Partial difference equations usually describe the evolution of certain phenomena over the course of time. Elementary but relevant examples of partial difference equations are concerned with heat diffusion, heat control, temperature distribution, population growth, cellular neural networks, etc. The interest for problems of this type is motivated by major applications of differential and difference operators to various applied fields, such as electrorheological (smart) fluids, space technology, robotics, image processing, etc. In this chapter, we describe several problems described by partial difference equations involving nonhomogeneous operators. The results contained in this chapter are based on the original contributions included in the papers [44, 85, 60, 61, 68, 73].