  These are extended notes of a talk given at Maurice Auslander Distinguished
Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim
is to give an introduction into Schubert calculus on Grassmannians and flag
varieties. We discuss various aspects of Schubert calculus, such as
applications to enumerative geometry, structure of the cohomology rings of
Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude
with a survey of results of V.Kiritchenko, V.Timorin and the author on a new
approach to Schubert calculus on full flag varieties via combinatorics of
Gelfand-Zetlin polytopes.
