Tris(2-benzimidazolylmethyl)amines have been found to be superior accelerating ligands for the copper(I)-catalyzed azide-alkyne cycloaddition reaction. Candidates bearing different benzimidazole N-substituents as well as benzothiazole and pyridyl ligand arms were evaluated by absolute rate measurements under relatively dilute conditions by aliquot quenching kinetics and by relative rate measurements under concentrated conditions by reaction calorimetry. Benzimidazole-based ligands with pendant alkylcarboxylate arms proved to be advantageous in the latter case. The catalyst system was shown to involve more than one active species, providing a complex response to changes in pH and buffer salts and the persistence of high catalytic rate in the presence of high concentrations of coordinating ligands. The water-soluble ligand (BimC4A)3 was found to be especially convenient for the rapid and high-yielding synthesis of several functionalized triazoles with 0.01-0.5 mol % Cu.