Schubert varieties in the flag variety *G/B*of a linear
algebraic group *G* over an algebraically closed field were originally
defined by Chevalley in his famous lost paper, where he also showed that
the singular locus of a Schubert variety has codimension at least two.
Perhaps unintentionally, he also remarked that Schubert varieties are
probably smooth. This talk will give a survey of the global smoothness
results for Schubert varieties in *G/B*. The results use a nice mixture of
the combinatorics of the Weyl group of *G* and the geometry of *G/B*.