  In this paper we consider various problems involving the action of a
reductive group $G$ on an affine variety $V$. We prove some general rationality
results about the $G$-orbits in $V$. In addition, we extend fundamental results
of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of
$G$ for such general $G$-actions.
  We apply our general rationality results to answer a question of Serre
concerning the behaviour of his notion of $G$-complete reducibility under
separable field extensions. Applications of our new optimality results also
include a construction which allows us to associate an optimal destabilizing
parabolic subgroup of $G$ to any subgroup of $G$. Finally, we use these new
optimality techniques to provide an answer to Tits' Centre Conjecture in a
special case.
