  In this survey paper we review recent advances in the calculus of
Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical
Schubert varieties. These three theories are one-parameter ($\hbar$)
deformations of the notion of fundamental class in their respective
extraordinary cohomology theories. Examining these three classes in conjunction
is justified by their relation to Okounkov's stable envelope notion. We review
formulas for the $\hbar$-deformed classes originating from Tarasov-Varchenko
weight functions, as well as their orthogonality relations. As a consequence,
explicit formulas are obtained for the Littlewood-Richardson type structure
constants.
