Let be a ring, be a left -module and . When is semidualizing, the Auslander class and the Bass class associated with have been the subject of extensive investigations. It has been proved that these classes, also known as Foxby classes, are one of the central concepts of (relative) Gorenstein homological algebra. In this paper, we answer several natural questions which arise when we weaken the condition of being semidualizing: if we let be w-tilting (see Definition 2.1), we establish the conditions for the pair to be a perfect cotorsion theory and for the pair to be a complete hereditary cotorsion theory. This tells us when the classes of Auslander and Bass are preenveloping and precovering, which generalizes a number of results disseminated in the literature. We investigate Gorenstein flat modules relative to a not necessarily semidualizing module and we find conditions for the class of -projective modules to be special precovering, the class of -flat modules to be covering, the one of Gorenstein -projective modules to be precovering and that of Gorenstein -injective modules to be preenveloping. We also find how to recover Foxby classes from -resolutions of .