Here are three large-scale issues. First, in what good sense or senses, if any, does ZFC provide a foundation for mathematics. Second, can category theory provide an alternative(?), or better(?) foundation for some or all mathematics? Third, should set theory should be understood as the study of a single universe, or as the study of a multiverse of set-theoretic universes? Each of these questions is many-faceted, and there are evident connections.
Penelope Maddy takes them all on in a substantial paper ‘Set-theoretic Foundations‘ (forthcoming in a festschrift for Woodin). She writes: “I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand.” Written with her customary lucidity, Maddy does indeed do some useful sorting out, it seems to me. Worth a read if you haven’t seen the paper yet.