Emergentism has aged into something close to the default, if often loosely stated, position of working neuroscientists and cognitive scientists, who routinely describe consciousness, or mind, as emerging from brain complexity without committing themselves to Broad's stronger, more precise claim that this emergence is unpredictable even in principle; this weaker, more common usage is sometimes distinguished as weak emergence, contrasted with the strong emergence Broad and his contemporaries actually defended.
In the mind-body debate specifically, emergentism sits close to epiphenomenalism in some formulations, since an emergent mental property might still be denied any causal power of its own over the physical base that produced it, while other emergentists insist on downward causation, mental properties genuinely acting back on the physical parts from which they arose, a stronger and more contested claim. The position's appeal outside strict analytic philosophy of mind is broad: it offers a way to speak of a genuinely new, higher-order reality, whether mind, society, or life itself, arising from physical process without either denying physicalism or positing an immaterial soul, which is exactly why it recurs across process theology and systems biology as much as in philosophy of mind proper.