Jaegwon Kim and others have claimed that (strong) psychophysical supervenience entails the reducibility of mental properties to physical properties. I argue that this claim is unwarranted with respect to epistemic (explanatory) reducibility (either of a ‘global’ or of a ‘local’ sort), as well as with respect to ontological reducibility. I then attempt to show that a robust version of nonreductive materialism (which I call ‘supervenient token-physicalism’) can be defended against the charge that nonreductive materialism leads to epiphenomenalism in failing to account for the causal or explanatory relevance of mental properties.