The objective of this chapter is the analytical presentation of the Z transform, which is the main tool for describing discrete-time systems in the frequency domain. Following a discussion that includes the defining equation of this transform and its relationship with the Laplace transform, poles and zeros are defined in the same way as in the continuous-time domain, and the conditions of convergence of this transform are presented. Then the focus shifts to the statement and proof of the properties of Z transforms, and the initial and final value theorems are presented. The inverse Z transform is then defined, and the three well-known methods of its calculation (namely, integration in the complex plane, power series expansion, and partial fraction expansion) are discussed. The chapter concludes with a presentation of the Z transform of the product of two discrete sequences, as well as the well-known Parseval identity. The unilateral (or one-sided) Z transform is also presented using the case of nonnegative values of discrete-time variables associated with causal discrete-time signals.