Supervenience' is now part of the philosophical vocabulary of the analytical tradition. Jaegwon Kim has a major responsibility for this. It is fair to say that he has done more than anyone else to identify types of supervenience, to examine their pairwise logical relationships, and to explore their usefulness for various philosophical purposes. As a body, his papers so thoroughly and carefully examine supervenience that, after reading them, one can be left with the feeling there is little worth saying about supervenience that he has not said. I shall, however, attempt to make a few worthwhile points about supervenience, and to look at supervenience in one context in which Kim has not.The paper is divided into two parts, each with subsections. In the first part, I shall discuss some matters that have been extensively examined by Kim, namely what the basic types of supervenience are and how they are pairwise logically related; in the course of this discussion, I shall distinguish a weak from a strong notion of global supervenience. In the second part, I shall examine supervenience in a context in which Kim has not: I shall attempt to solve a puzzle that arises when we consider supervenience relations involving vague properties and/or predicates. This is, of course, not a special case of property/predicate supervenience, but the typical one: for virtually all properties/predicates are vague. 2 The two parts of the paper stand independently of each other. Of the discussion in the first part, essentially all that is presupposed in the second is the notion of "world strong supervenience" defined in section 1 of part I.