We establish the formula for multiplication by the class of a special
Schubert variety in the integral cohomology ring of the flag manifold.
This formula also describes the multiplication of a Schubert polynomi
al by either an elementary or a complete symmetric polynomial. Thus, w
e generalize the classical Pieri's formula for Schur polynomials (asso
ciated to Grassmann varieties) to Schubert polynomials (associated to
flag manifolds). Our primary technique is an explicit geometric descri
ption of certain intersections of Schubert varieties. This method allo
ws us to compute additional structure constants for the cohomology rin
g, some of which we express in terms of paths in the Bruhat order on t
he symmetric group, which in turn yields an enumerative result about t
he Bruhat order.