This thesis concentrates on the fundamental theory for discrete Volterra equations, or implicit Volterra difference equations with unbounded order, which can be considered as the analogue of Volterra integral equations. For this purpose, we develop the basic theory for discrete Volterra operators and establish several fixed point theorems for discrete Volterra operators by using Schauder's fixed point theorems and the contraction mapping theorem. As a direct application of the fixed point theorems for discrete Volterra operators, we establish several corresponding existence theorems for the solutions of discrete Volterra equations. We use the.

-transform techniques to studylinear discrete Volterra equations of convolution type, give a rigorous proof of discrete Paley-Wiener theorem, and discuss the structure of solutions. For linear Volterra equations of non-convolution type, the corresponding basic theory is also developed. In addition, we develop the stability and perturbation theory for discrete Volterra equations and study periodic solutions of discrete Volterra equations. Finally, for a class of Volterra integral and integro-differential equations having convolution kernels with fading memory, we provide an efficient method to investigate the long-term qualitative behavior of the corresponding numerical solutions. The work in this thesis provides a foundation for further study of discretized versions of Volterra integral and integro-differential equations.