The cohomology rings of flag manifolds, with their rich combinatorial structure, have been vital in many areas of mathematics, dating back to the classical Schubert calculus of enumerative geometry. They continue to be a source of new insights, structures, and applications. In this talk, I will discuss some recent advances in the understanding of these rings, particularly concrete aspects of the interplay between the geometry and combinatorics of flag manifolds. I will focus on Pieri-type formulas for products of Schubert classes.