This chapter introduces the Z transform as a key tool for analyzing discrete-time signals and systems, with broad applications in digital signal processing and control systems. It begins with an overview of the Z transform’s historical development and definition, followed by its core mathematical properties, including linearity, translation, multiplication, and convolution. Essential theorems such as Parseval’s and the initial and final value theorems and techniques for handling partial derivatives in the Z-domain are covered. A detailed discussion of the inverse Z transform enables the reconstruction of original signals. Practical uses, like solving difference equations and defining transfer functions, are demonstrated, with a focus on system analysis through poles and zeros. The chapter also covers frequency response and the role of pole-zero configurations in assessing system stability and summing infinite series.