  We derive N = 1, 2 superfield equations as the conditions for a (nonlinear)
theory of one abelian N = 1 or N = 2 vector multiplet to be duality invariant.
The N = 1 super Born-Infeld action is a particular solution of the
corresponding equation. A family of duality invariant nonlinear N = 1
supersymmetric theories is described. We present the solution of the N = 2
duality equation which reduces to the N = 1 Born-Infeld action when the (0,1/2)
part of N = 2 vector multiplet is switched off. We also propose a constructive
perturbative scheme to compute duality invariant N = 2 superconformal actions.
