  Starting from a recently proposed linear formulation in terms of auxiliary
fields, we study $n$-field generalizations of Born and Born-Infeld theories. In
this description the Lagrangian is quadratic in the vector field strengths and
the symmetry properties (including the characteristic self-duality) of the
corresponding non-linear theory are manifest as on-shell duality symmetries and
depend on the choice of the (homogeneous) manifold spanned by the auxiliary
scalar fields and the symplectic frame. By suitably choosing these defining
properties of the quadratic Lagrangian, we are able to reproduce some known
multi-field Born-Infeld theories and to derive new non-linear models, such as
the $n$-field Born theory. We also discuss non-Abelian generalizations of these
theories obtained by choosing the vector fields in the adjoint representation
of an off-shell compact global symmetry group $K$ and replacing them by
non-Abelian, $K$-covariant field strengths, thus promoting $K$ to a gauge
group.
