  We give upper bounds on the essential dimension of (quasi-)simple algebraic
groups over an algebraically closed field that hold in all characteristics. The
results depend on showing that certain representations are generically free. In
particular, aside from the cases of spin and half-spin groups, we prove that
the essential dimension of a simple algebraic group $G$ of rank at least two is
at most $\mathrm{dim}(G) - 2(\mathrm{rank}(G)) - 1$. It is known that the
essential dimension of spin and half-spin groups grows exponentially in the
rank. In most cases, our bounds are as good or better than those known in
characteristic zero and the proofs are shorter. We also compute the generic
stabilizer of an adjoint group on its Lie algebra.
