This article tries to fill a descriptive gap in the literature on English possessive NPs by offering a systematic description of indefinite possessive NPs (e.g. a friend's house, a widow's pension, men's suits). Specifically, two descriptive issues are dealt with. Firstly, the different types of determining indefinite genitives are discussed. It is shown that indefinite determiner genitives can have specific reference, generalized reference or generic reference. Moreover, it is argued that in all of these cases, the indefinite genitive specifies a reference point (Langacker 1993) for the identification of the ‘possessee’ and functions as a grounding predication. It is also proposed that subsective gradience (Aarts 2004, 2007) can be diagnosed for determining genitives, i.e. that more central and more peripheral members can be distinguished within this category. Secondly, the distinction between determining and nondetermining, classifying genitives (e.g. a [woman's magazine] ‘a magazine for women’) is re-examined. This distinction is relevant to the description of indefinite possessive NPs, since they have been considered as evidence for a fuzzy boundary between determining and nondetermining genitives (Taylor 1996, Rosenbach 2006). It is argued, against such an analysis, that the two functions of the genitive can be clearly distinguished, even in ‘bridging contexts’ (Evans & Wilkins 2000), i.e. contexts in which both a determining and a nondetermining reading are supported. Consequently, there is no intersective gradience (Aarts 2004, 2007) between determining and nondetermining genitives on the level of the constructional semantics.