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We introduce and develop an analogous of the Auslander–Buchweitz approximation theory (see Auslander and Buchweitz, Societe Mathematique de France 38:5–37, 1989) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category equipped with a pair where is closed under extensions and ω is a weak-cogenerator in usually under additional conditions. This reduces …