  We extend the results of Cellini-Papi on the characterizations of nilpotent
and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple
Lie algebra. These characterizations are given in terms of elements of the
affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra
of a classical Lie algebra, we give formulas for the number of these ideals.
