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We prove that an algebraic stack with affine stabilizers over an arbitrary(possibly mixed-characteristic) base is \'etale locally a quotient stack aroundany point with a linearly reductive stabilizer. This generalizes earlier workby the authors (stacks over algebraically closed fields) and by Abramovich,Olsson and Vistoli (stacks with finite inertia). In addition, we prove a numberof foundational results, which are new even over a field. We expect theseresults to be as consequential to the study of algebraic stacks as our localstructure theorems. They include various coherent completeness and effectivityresults for adic sequences of algebraic stacks. Finally, we give severalapplications of our results and methods, such as structure theorems forlinearly reductive group schemes and generalizations to the relative setting ofSumihiro's theorem on torus actions and Luna's \'etale slice theorem.

The \'etale local structure of algebraic stacks