Suppose that a group acts on a variety. When can the variety and the action be resolved so that all stabilizers are finite and the space of orbits desingularized, at least partially? Kirwan gave an answer to this question in the 1980s through an explicit blowup algorithm for smooth varieties with group actions in the context of Geometric Invariant Theory (GIT). In this talk, we will explain how to generalize Kirwan's algorithm to Artin stacks and their moduli spaces in derived algebraic geometry, which, in particular, include classical, potentially singular, quotient stacks that arise from group actions in GIT. Based on joint works with (subsets of) Eric Ahlqvist, Jeroen Hekking, Michele Pernice and David Rydh.