Supervenience is a topic-neutral, broadly logical relation between classes of properties or facts. In a slogan, A supervenes on B if and only if there cannot be an A-difference without a B-difference. The first part of this paper considers different ways in which that slogan has been cashed out. The second part discusses applications of concepts of supervenience, focussing on the question whether they may provide an explication of determination theses such as physicalism.