Existence of moduli spaces for algebraic stacks

We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a $\mathrm{\Theta }$-stratification. These results provide a generalization of the Keel–Mori theorem to moduli problems whose objects have positive dimensional automorphism groups and give criteria on the moduli problem to have a separated or proper good moduli space. To illustrate our method, we apply these results to construct proper moduli spaces parameterizing semistable $\mathcal{G}$-bundles on curves and moduli spaces for objects in abelian categories.