  A theorem of Ryan and Wolper states that a type A Schubert variety is smooth
if and only if it is an iterated fibre bundle of Grassmannians. We extend this
theorem to arbitrary finite type, showing that a Schubert variety in a
generalized flag variety is rationally smooth if and only if it is an iterated
fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof
depends on deep combinatorial results of Billey-Postnikov on Weyl groups. We
determine all smooth and rationally smooth Grassmannian Schubert varieties, and
give a new proof of Peterson's theorem that all simply-laced rationally smooth
Schubert varieties are smooth. Taken together, our results give a fairly
complete geometric description of smooth and rationally smooth Schubert
varieties using primarily combinatorial methods.
