N=1 and N=2 supersymmetric non-abelian Born–Infeld actions from superspace

New non-abelian supersymmetric generalizations of the four-dimensional Born–Infeld action are constructed in N=1 and N=2 superspace, to all orders in α′. The proposed actions are dictated by simple (manifestly supersymmetric and gauge-covariant) non-linear constraints.