We report on recent progress in the field of Schubert calculus, a classical branch of enumerative geometry, due to its surprising connection to quantum integrable systems. We shall see how the latter provide many explicit combinatorial formulae (``puzzle rules'') for intersection numbers for partial flag varieties, and their generalizations (e.g. in equivariant K-theory). We shall also discuss the connection with the work of Okounkov et al on quantum integrable systems and the equivariant cohomology of Nakajima quiver varieties. This is joint work with A. Knutson (Cornell).