  Hilbert's 15th problem called for a rigorous foundation of Schubert's
calculus, in which a long standing and challenging part is Schubert's problem
of characteristics. In the course of securing the foundation of algebraic
geometry, Van der Waerden and Andr\'{e} Weil attributed the problem to the
determination of the intersection theory of flag manifolds.
  This article surveys the background, content, and resolution of the problem
of characteristics. Our main results are a unified formula for the
characteristics, and a system description for the intersection rings of flag
manifolds. We illustrate the effectiveness of the formula and the algorithm via
explicit examples.
