The idea of a higher level phenomenon having a downward causal influence on a lower level process or entity has taken a variety of forms. In order to discuss the relation between emergence and downward causation, the specific variety of the thesis of downward causation (DC) must be identified. Based on some ontological theses about inter-level relations, types of causation and the possibility of reduction, three versions of DC are distinguished. Of these, the `Strong' form of DC is held to (...) be in conflict with contemporary science; the `Medium' version of DC may for instance describe thoughts constraining neurophysiological states, while the `Weak' form of DC is physically acceptable but may not in practice be a feasible description of the mind/brain or the cell/molecule relation. All forms have their specific problems, but the Medium and the Weak version seems to be most promising. (shrink)