Starting from the assumption that NPs of all kinds can be anaphoric on antecedents in the linguistic context, we work towards a general theory of context-dependent NP meaning. Two complicating factors are that the relation between anaphors and antecedents is by no means unrestricted and that often there is a partial match between anaphor and antecedent. We argue that the presuppositions-as-anaphors approach of van der Sandt provides a natural starting point for our enterprise. Unfortunately, this theory has a number of deficiencies for our purposes, in particular where the treatment of partial matches is concerned. We propose a number of modifications of van der Sandt's formal theory and apply the modified algorithm first to definite NPs and later to NPs of all kinds. The resulting modified version of the presuppositions-as-anaphors theory is argued to be more general, formally more precise, and empirically more adequate than its predecessor.