  We shown that, if you have two planes in the Segal-Wilson Grassmannian that
have an intersection of finite codimension, then the corresponding solutions of
the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The
pseudodifferential operator that performs this transformation is shown to be
built up in a geometric way from elementary BDT's and is given here in a closed
form. The geometric description of elementary DBT's requires that one has a
geometric interpretation of the dual wavefunctions involved. This is done here
with the help of a suitable algebraic characterization of the wavefunction. The
BDT's also induce transformations of the tau-function associated to a plane in
the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric
characterization of the BDT'ss that preserves these subsystems of the KP
hierarchy. This generalizes the classical Darboux-transformations. we also
determine an explicit expression for the squared eigenfunction potentials. Next
a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy.
It is shown that infinite flags in the Grassmannian yield solutions of the
latter hierarchy. these flags can be constructed by means of BDT's, starting
from some plane. Other applications of these BDT's are a geometric way to
characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy
and the construction of a vast collection of orthogonal polynomials, playing a
role in matrix models.
