This talk will focus on a geometric construction called compactified configuration spaces, where the resolution of point collisions can be seen as a real analogue of the Deligne--Mumford compactification of the moduli spaces of marked Riemann surfaces. I will talk about the local "bubbling" behavior, which globally has an iterative nature and is encoded in a combinatorial "tree" structure. This new construction has been an essential tool in the study of constant curvature conical metrics, and we expect it to provide a good framework for analyzing other problems involving families of elliptic operators with merging regular singularities. Joint work with Rafe Mazzeo.