  We show that the Born-Infeld theory with n complex abelian gauge fields
written in an auxiliary field formulation has a U(n,n) duality group. We
conjecture the form of the Lagrangian obtained by eliminating the auxiliary
fields and then introduce a new reality structure leading to a Born-Infeld
theory with n real fields and an Sp(2n,R) duality symmetry. The real and
complex constructions are extended to arbitrary even dimensions. The maximal
noncompact duality group is U(n,n) for complex fields. For real fields the
duality group is Sp(2n,R) if half of the dimension of space-time is even and
O(n,n) if it is odd. We also discuss duality under the maximal compact
subgroup, which is the self-duality group of the theory obtained by fixing the
expectation value of a scalar field. Supersymmetric versions of self-dual
theories in four dimensions are also discussed.
