Nil-Hecke rings and Schubert calculus

The purpose of the present notes is to give a self-contained exposition on the use of the techniques of Nil-Hecke algebras in the localization approach to the equivariant Schubert calculus for cohomology of flag varieties. We also demonstrate how these techniques can be applied to non-crystallographic root systems as well as to study (connective) $K$-theory of flag varieties.