  We introduce and study a class of algebraic stacks with finite inertia in
positive and mixed characteristic, which we call tame algebraic stacks. They
include tame Deligne-Mumford stacks, and are arguably better behaved than
general Deligne-Mumford stacks. We also give a complete characterization of
finite flat linearly reductive schemes over an arbitrary base. Our main result
is that tame algebraic stacks are \'etale locally quotient by actions of
linearly reductive finite group schemes.
  In a subsequent paper we will show that tame algebraic stacks admit a good
theory of stable maps.
