Projective and injective modules are of key importance in algebra, in part because of their useful homological properties. The notion of C-projective and C-injective modules generalizes these constructions. In particular, these modules may be used to construct resolutions and define related homological dimensions in a natural way. When C is a semidualizing module, the C-projective and C-injective modules have particularly useful homological properties. Further, one may combine projective and C-projective resolutions to construct complete PC-resolutions (and, dually, complete IC-resolutions) that yield other modules with nice homological properties. This paper surveys some of the literature on these constructions.