  Let k be a separably closed field. Let G be a reductive algebraic k-group. In
this paper, we study Serre's notion of complete reducibility of subgroups of G
over k. In particular, using the recently proved center conjecture of Tits, we
show that the centralizer of a k-subgroup H of G is G-completely reducible over
k if it is reductive and H is G-completely reducible over k. We also show that
a regular reductive k-subgroup of G is G-completely reducible over k. Various
open problems concerning complete reducibility are discussed. We present
examples where the number of overgroups of irreducible subgroups and the number
of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.
