  We classify all possible automorphism groups of smooth cubic surfaces over an
algebraically closed field of arbitrary characteristic. As an intermediate step
we also classify automorphism groups of quartic del Pezzo surfaces. We show
that the moduli space of smooth cubic surfaces is rational in every
characteristic, determine the dimensions of the strata admitting each possible
isomorphism class of automorphism group, and find explicit normal forms in each
case. Finally, we completely characterize when a smooth cubic surface in
positive characteristic, together with a group action, can be lifted to
characteristic zero.
