  We construct Koppelman formulas on manifolds of flags in $\C^N$ for forms
with values in any holomorphic line bundle as well as in the tautological
vector bundles and their duals. As an application we obtain new explicit proofs
of some vanishing theorems of the Bott-Borel-Weil type by solving the
corresponding $\debar$-equation. We also construct reproducing kernels for
harmonic $(p,q)$-forms in the case of Grassmannians.
