Alper, Jarod2015-12-080022-4049http://hdl.handle.net/1885/33843We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds �tale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space, generalizing the results of Pinkham and Rim. We provide a generalization and stack-theoretic proof of Luna�s �tale slice theorem which shows that GIT quotient stacks are �tale locally quotients stacks by the stabilizer.201010.1016/j.jpaa.2009.11.0162016-06-14