Abstract
Using superspace techniques, the complete and most general action of D = 10 super--Yang-Mills theory is constructed at the α'2 level. No other approximations, e.g., keeping only a subset of the allowed derivative terms, are used. The Lorentz structure of the α'2 corrections is completely determined, while (depending on the gauge group) there is some freedom in the adjoint structure, which is given by a totally symmetric four-index tensor. We examine the second, non-linearly realised supersymmetry that may be present when the gauge group has a U(1) factor, and find that the constraints from linear and non-linear supersymmetry to a large extent coincide. However, the additional restrictions on the adjoint structure of the order α'2 interactions following from the requirement of non-linear supersymmetry do not completely specify the symmetrised trace prescription.