The aim of this paper is to define a notion of supervenience which can adequately describe the systematic dependence of extrinsic as well as of intrinsic higher-level properties on base-level features. We argue that none of the standard notions of supervenience—the concepts of weak, strong and global supervenience—fulfil this function. The concept of regional supervenience, which is purported to improve on the standard conceptions, turns out to be problematic as well. As a new approach, we develop the notion of property-dependent supervenience. This notion is founded on a criterion of relevance adapting the supervenience base to the considered higher-level properties in a specific way, such that only features which are relevant to the instantiation of the higher-level properties under consideration are taken into account.