The copper(I)-catalyzed azide-alkyne cycloaddition (CuAAC) was discovered in 2002, which has become the most remarkable example for "click chemistry" to date. In CuAAC reaction, 1-copper(I) alkyne has been recognized to be a key intermediate. However, many contradictory experimental results for this intermediate were reported in literature. For example, only the in-situ generated 1-copper(I) alkyne was used, while the premade 1-copper(I) alkyne proved to be inefficient under the standard conditions. The kinetic studies indicated that CuAAC reaction had a strict second-order dependence on Cu(I) and the DFT studies demonstrated that 1-copper(I) alkyne intermediate should be a dinuclear copper(I) complex. But these results were inconsistent with the structure of the premade 1-copper(I) alkyne. Although hundreds of structurally different ligands were reported to significantly enhance the efficiency of CuAAC reaction, their functions were assigned to prevent the oxidation and the disproportionation of Cu(I) ion. Based on the investigation of the references and our works, we proposed that the in-situ generated 1-copper(I) alkyne in CuAAC reaction is not identical with the premade 1-copper(I) alkyne. The ligands may play dual roles to activate the 1-copper(I) alkyne by blocking the polymerization of the in-situ formed 1-copper(I) alkynes and dissociating the polymeric structures of the premade 1-copper(I) alkynes. As a result, we first disclosed that carboxylic acids can function as such activators and a novel carboxylic acid-catalyzed CuAAC strategy was developed, which has been proven to be the most convenient and highly efficient CuAAC method to date. Furthermore, highly efficient and regioselective methods for the syntheses of 1,4,5-trisubstituted 1,2,3-triazoles were developed by using the premade 1-copper(I) alkynes as substrates, in which the novel function of the premade 1-copper(I) alkynes as excellent dipolarophiles was first disclosed and applied. In this article, a series of works reported by our group for the in-situ generated and the premade 1-copper(I) alkynes in cycloadditions are reviewed.