This paper gives an overview of the stability analysis of systems with delay-dependent coefficients. Such systems are frequently encountered in various scientific and engineering applications. Most such analyses are generalization of those on systems with delay-independent coefficients. Therefore an introduction on systems with delay-independent coefficients is also given, with an emphasis on the so-called τ-decomposition approach. Methods for two key ingredients of this approach are discussed, namely the identification of imaginary characteristic roots with the corresponding delays, and local behavior analysis of these roots as the delay, seen as parameters, increases through these critical values. For systems with delay-dependent coefficients, we review the analysis methods for systems with a single delay and commensurate delays, their application to output feedback control as well as a geometric perspective that establishes a link between systems with or without delay-dependent coefficients. We provide the main ideas of various stability analysis methods and their advantages and limitations. We also present our perspectives on future directions of research on this interesting topic.