  Let G be a simply-connected simple compact Lie group over the complex
numbers. The affine Grassmannian is a projective ind-variety,
homotopy-equivalent to the loop space of G and closely analogous to a maximal
flag variety of the classical Grassmannian manifold. It has a Schubert cell
decomposition indexed by the coroot lattice or equivalently by the minimal
length coset representatives for the affine Weyl group modulo the Weyl group
for G. The closure of an affine Schubert cell is a finite dimensional
projective variety that we call an affine Schubert variety. In this paper we
completely determine the smooth and palindromic (rationally smooth) affine
Schubert varieties.
