The copper(I)-catalyzed azide-alkyne cycloaddition (CuAAC) reaction regiospecifically produces 1,4-disubstituted-1,2,3-triazole molecules. This heterocycle formation chemistry has high tolerance to reaction conditions and substrate structures. Therefore, it has been practiced not only within, but also far beyond the area of heterocyclic chemistry. Herein, the mechanistic understanding of CuAAC is summarized, with a particular emphasis on the significance of copper/azide interactions. Our analysis concludes that the formation of the azide/copper(I) acetylide complex in the early stage of the reaction dictates the reaction rate. The subsequent triazole ring-formation step is fast and consequently possibly kinetically invisible. Therefore, structures of substrates and copper catalysts, as well as other reaction variables that are conducive to the formation of the copper/alkyne/azide ternary complex predisposed for cycloaddition would result in highly efficient CuAAC reactions. Specifically, terminal alkynes with relatively low pKa values and an inclination to engage in π-backbonding with copper(I), azides with ancillary copper-binding ligands (aka chelating azides), and copper catalysts that resist aggregation, balance redox activity with Lewis acidity, and allow for dinuclear cooperative catalysis are favored in CuAAC reactions. Brief discussions on the mechanistic aspects of internal alkyne-involved CuAAC reactions are also included, based on the relatively limited data that are available at this point.