I outline reasons for the recent popularity, and lingering suspicion, about 'emergence' by examining three distinct concepts of property emergence, their purposes and associated obligations. In Part 1, I argue 'Strong' emergence is the grail for many emergentists (and physicalists), since it frames what is needed to block the 'Argument from Realization' (AR) which moves from the truth of physicalism to the inefficacy of special science properties. I then distinguish 'Weak' and 'Ontological' emergence, in Part 2, arguing each is a way one may fail to establish the possibility of Strong emergence. But I also show Weak emergence can help the full-blown reductionist and Ontological emergence helps those opposed to physicalism. Lastly, in Part 3, I argue that the Completeness of Physics (CoP) is incompatible with Strong emergence and that rejecting CoP provides hope for the possibility of Strong emergence in a physical world. The result is a notion of Strong emergence offering much to non-reductive physicalism. My final conclusion is that concepts of emergence, when properly understood, have important contributions to make to philosophical debate.