  The concept of self-dual supersymmetric nonlinear electrodynamics is
generalized to a curved superspace of N = 1 supergravity, for both the old
minimal and the new minimal versions of N = 1 supergravity. We derive the
self-duality equation, which has to be satisfied by the action functional of
any U(1) duality invariant model of a massless vector multiplet, and construct
a family of self-dual nonlinear models. This family includes a curved
superspace extension of the N = 1 super Born-Infeld action. The supercurrent
and supertrace in such models are proved to be duality invariant. The most
interesting and unexpected result is that the requirement of nonlinear
self-duality yields nontrivial couplings of the vector multiplet to Kahler
sigma models. We explicitly derive the couplings to general Kahler sigma models
in the case when the matter chiral multiplets are inert under the duality
rotations, and more specifically to the dilaton-axion chiral multiplet when the
group of duality rotations is enhanced to SL(2,R).
