This chapter covers the most important techniques related to Fourier analysis that are applied to discrete-time signals and systems. The chapter begins with a presentation of Fourier series of discrete-time signals, its properties, and its physical significance, followed by the fundamental theory of discrete-time Fourier transform (DTFT) and related material, such as the conditions of its convergence, its relationship to the \(\mathcal {Z}\) transform, its definition for the case of periodic signals, and its main properties. The inverse DTFT is also discussed. The focus then shifts to a frequency analysis of discrete-time systems and a presentation of concepts such as frequency selective filters and geometric estimation of the frequency response function. To present the theory of the fundamental discrete Fourier transform (DFT), the technique of sampling in the frequency domain is first discussed, followed by a detailed description of the DFT, its properties, its efficient estimation via the powerful fast Fourier transform (FFT), and the relationship between linear and circular convolution used with discrete sequences of finite length. The chapter concludes with an analysis of short-time signals, more specifically, with a presentation of the time-dependent Fourier transform (TDFT) and the method of the so-called overlap-add reconstruction technique.