Non-reductive physicalists have made a number of attempts to provide the relation of supervenience between levels of properties with enough bite to analyze interesting cases without at the same time losing the relation's acceptability for the physicalist. I criticize some of these proposals and suggest an alternative supplementation of the supervenience relation by imposing a requirement of robustness which is motivated by the notion of structural stability familiar from dynamical systems theory. Robust supervenience, I argue, captures what the non-reductive physicalist (...) wants from supervenience; most importantly, it provides a natural background for reconstructing the notion of (diachronic) property emergence in a way acceptable to physicalists. (shrink)