A couple weeks ago I attended a talk about the Keel-Mori (and Conrad?) theorem regarding existence of coarse moduli spaces for Deligne-Mumford stacks with finite inertia. Here are some questions that I have been wondering about since then: What are some applications of this theorem? What does it matter if a DM stack has a coarse space? What are examples of things that we can do with the coarse space that we maybe can't do with the stack? Given (for instance) a moduli problem, what does the existence of a coarse moduli space tell us that the existence of (say) a DM moduli stack doesn't necessarily tell us?