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