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 polynomial by either
an elementary or a complete symmetric polynomial. Thus, we generalize the
classical Pieri's formula for symmetric polynomials/Grassmann varieties to
Schubert polynomials/flag manifolds. Our primary technique is an explicit
geometric description of certain intersections of Schubert varieties. We
compute additional structure constants for the cohomology ring, some of
which we express in terms of paths in the Bruhat order on the symmetric
group and obtain an enumerative result about the Bruhat order.