  In this paper, we study the index for several natural classes of
non-reductive subalgebras of semisimple Lie algebras. Namely, we look at
parabolic subalgebras, centralisers of nilpotent elements, and the normalisers
of the centralisers. We discuss a conjecture of Elashvili to the effect that
the index of any centraliser is equal to the rank of the semisimple algebra in
question. It is shown that Elashvili's conjecture is true for `small' and
`large' orbits. Some properties of the index for the normaliser of the
centraliser are proved. In particular, we prove that, for a regular nilpotent
element, the normaliser of the centraliser is a Frobenius Lie algebra.
