  Extriangulated categories were introduced by Nakaoka and Palu as a
simultaneous generalization of exact categories and triangulated categories. In
this paper, we introduce and develop an analogous theory of Auslander-Buchweitz
approximations for extriangulated categories. We establish the existence of
precovers pand preenvelopesq and obtain characterizations of relative
homological dimensions, which are based on certain subcategories under
finiteness of resolutions. Finally, we give a description of cotorsion pairs on
extriangulated categories under some conditions, and provide a characterization
of silting subcategories on stable categories. Keywords: Extriangulated
category; Homological dimension; Cogenerator; Cotorsion pair.
