Second flip in the Hassett-Keel program: existence of good moduli spaces

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel{Maori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne-Mumford stack. We apply this result to prove that the moduli stacks (M) over bar (g,n) (alpha) parameterizing alpha-stable curves introduced in [J. Alper et al., Second flip in the Hassett-Keel program: a local description, Compositio Math. 153 (2017), 1547-1583] admit good moduli spaces.