This chapter presents the basic theory of discrete-time systems. It begins with a presentation of the types and categories of discrete-time systems. These systems, like their continuous-time counterparts, can also be classified as static or dynamic, time-varying or time-invariant, linear or nonlinear, stable or unstable, and causal or noncausal. The concepts of impulse response, step response, and frequency response are then defined and calculated for the special case of linear time-invariant (LTI) discrete-time systems. The special case of causal and stable systems is then presented together with the interesting category of recursive systems. The system interconnection that allows for the combinations of simple systems into complex higher-order systems is then analyzed, and the concept of the inverse system is discussed, together with the related concepts of direct form I and direct form II system representations. The chapter concludes with a description of discrete-time systems in state space, which includes the derivation and solution of the relevant dynamic equations, as well as the formulation of the most important characteristics of those systems in state space. Finally, so-called similarity transformations, together with the concepts of controllability, observability, and stability in state space, are presented.