The classic click reaction is the copper-catalyzed reaction of an azide with an alkyne to form a 5-membered heteroatom ring: a Cu(I)-catalyzed azide-alkyne cycloaddition (CuAAC). The first triazole synthesis, from diethyl acetylenedicarboxylate and phenyl azide, was reported by Arthur Michael in 1893. Later, in the middle of the 20th century, this family of 1,3-dipolar cycloadditions took on Rolf Huisgen's name after his studies of their reaction kinetics and conditions.
The copper(I)-catalysis of the Huisgen 1,3-dipolar cycloaddition was discovered concurrently and independently by the groups of Valery V. Fokin and K. Barry Sharpless at the Scripps Research Institute in California and Morten Meldal in the Carlsberg Laboratory, Denmark. The copper-catalyzed version of this reaction gives only the 1,4-isomer, whereas Huisgen's non-catalyzed 1,3-dipolar cycloaddition gives both the 1,4- and 1,5-isomers, is slow, and requires a temperature of 100 degrees Celsius.
Moreover, this copper-catalyzed "click" does not require ligands on the metal, although accelerating ligands such as tris(triazolyl)methyl amine ligands with various substituents have been reported and used with success in aqueous solution. Other ligands such as PPh3 and TBIA can also be used, even though PPh is liable to Staudinger ligation with the azide substituent. CuO in water at room temperature was found also to catalyze the same reaction in 15 minutes with 91% yield.
The first reaction mechanism proposed included one catalytic copper atom; but isotope, kinetic, and other studies have suggested a dicopper mechanism may be more relevant. Even though this reaction proceeds effectively at biological conditions, copper in this range of dosage is cytotoxic. Solutions to this problem have been presented, such as using water-soluble ligands on the copper to enhance cell penetration of the catalyst and thereby reduce the dosage needed, or to use chelating ligands to further increase the effective concentration of Cu(I) and thereby decreasing the actual dosage.
Although the Cu(I)-catalyzed variant was first reported by Meldal and co-workers for the synthesis of peptidotriazoles on solid support, their conditions were far from the true spirit of click chemistry  and were overtaken by the publicly more recognized Sharpless. Meldal and co-workers also chose not to label this reaction type "click chemistry" which allegedly caused their discovery to be largely overlooked by the mainstream chemical society. Fokin and Sharpless independently described it as a reliable catalytic process offering "an unprecedented level of selectivity, reliability, and scope for those organic synthesis endeavors which depend on the creation of covalent links between diverse building blocks".
An analogous RuAAC reaction catalyzed by ruthenium, instead of copper, was reported by the Jia and Fokin groups in 2005, and allows for the selective production of 1,5-isomers.