The previous chapter introduced the field of time-frequency (t, f) signal analysis and processing (TFSAP), which concerns the analysis and processing of signals with time-varying frequency content. Such signals are best represented by a time-frequency distribution (TFD), which is intended to show how the energy of the signal is distributed over the two-dimensional (t, f) space. Processing of the signal may then exploit the features produced by the concentration of signal energy in two dimensions (time and frequency) instead of only one (time or frequency). This chapter begins with the more formal, in-depth part of the tutorial which occupies the rest of Part I (Chapters 1–3). The present tutorial updates the one given in Ref. [1] and the first edition of this book by including more recent advances, refining terminology, and simplifying both the presentations of concepts and formulations of methodologies. This core material needs to be understood by anybody seeking to specialize in this field. Precise definitions and formulations of the key concepts needed to formulate (t, f) methods are provided. Chapter 1 includes four sections. Section 1.1 describes in detail the justifications why timefrequency methods are preferred for a wide range of applications in which the signals have time-varying spectral characteristics or multiple components for which the variables t and f are related. Section 1.2 provides the signal models and mathematical formulations needed to describe temporal and spectral characteristics of nonstationary signals in the time-frequency domain. It defines such basic concepts as analytic signals, Hilbert transform (HT), bandwidth-duration product, and asymptotic signals. Section 1.3 defines the key quantities related to time-frequency methods, including the instantaneous frequency (IF), spectral delay (SD), and group delay (GD). Section 1.4 reinforces the material in the previous sections with a simple tutorial on defining the AM/FM characteristics of signals using the concepts of analytic signal and IF; and relating to the definition of the amplitude and phase of a nonstationary signal.