  The main topic of the paper is the construction of various explicit flat
families of border bases. To begin with, we cover the punctual Hilbert scheme
Hilb^\mu(A^n) by border basis schemes and work out the base changes. This
enables us to control flat families obtained by linear changes of coordinates.
Next we provide an explicit construction of the principal component of the
border basis scheme, and we use it to find flat families of maximal dimension
at each radical point. Finally, we connect radical points to each other and to
the monomial point via explicit flat families on the principal component.
