Several theories of emergence will be distinguished. In particular, these are synchronic, diachronic, and weak versions of emergence. While the weaker theories are compatible with property reductionism, synchronic emergentism and strong versions of diachronic emergentism are not. Synchronice mergentism is of particular interest for the discussion of downward causation. For such a theory, a system's property is taken to be emergent if it is irreducible, i.e., if it is not reductively explainable. Furthermore, we have to distinguish two different types of (...) irreducibility with quite different consequences: If, on the one hand, a system's property is irreducible because of the irreducibility of the system's parts' behavior on which the property supervenes, we seem to have a case of "downward causation". This kind of downward causation does not violate the principle of the causal closure of the physical domain. If, on the other hand, a systemic property is irreducible because it is not exhaustively analyzable in terms of its causal role, downward causation is not implied. Rather, it is dubitable how unanalyzable properties might play any causal role at all. Thus, epiphenomenalism seems to be implied. The failure to keep apart the two kinds of irreducibility has muddled recent debate about the emergence of properties considerably. (shrink)