  We obtain a geometric construction of a ``standard monomial basis'' for the
homogeneous coordinate ring associated with any ample line bundle on any flag
variety. This basis is compatible with Schubert varieties, opposite Schubert
varieties, and unions of intersections of these varieties. Our approach relies
on vanishing theorems and a degeneration of the diagonal ; it also yields a
standard monomial basis for the multi-homogeneous coordinate rings of flag
varieties of classical type.
