  Here we study the problem of generalizing one of the main tools of Groebner
basis theory, namely the flat deformation to the leading term ideal, to the
border basis setting. After showing that the straightforward approach based on
the deformation to the degree form ideal works only under additional
hypotheses, we introduce border basis schemes and universal border basis
families. With their help the problem can be rephrased as the search for a
certain rational curve on a border basis scheme. We construct the system of
generators of the vanishing ideal of the border basis scheme in different ways
and study the question of how to minimalize it. For homogeneous ideals, we also
introduce a homogeneous border basis scheme and prove that it is an affine
space in certain cases. In these cases it is then easy to write down the
desired deformations explicitly.
