Copper-catalyzed azide-alkyne cycloaddition (CuAAC) is a widely utilized, reliable, and straightforward way for making covalent connections between building blocks containing various functional groups. It has been used in organic synthesis, medicinal chemistry, surface and polymer chemistry, and bioconjugation applications. Despite the apparent simplicity of the reaction, its mechanism involves multiple reversible steps involving coordination complexes of copper(I) acetylides of varying nuclearity. Understanding and controlling these equilibria is of paramount importance for channeling the reaction into the productive catalytic cycle. This tutorial review examines the history of the development of the CuAAC reaction, its key mechanistic aspects, and highlights the features that make it useful to practitioners in different fields of chemical science.