  We study the evolution of the flavour non-singlet deep-inelastic structure
functions F_{2,NS} and F_3 at the next-to-next-to-next-to-leading order (N^3LO)
of massless perturbative QCD. The present information on the corresponding
three-loop coefficient functions is used to derive approximate expressions of
these quantities which prove completely sufficient for values x > 10^{-2} of
the Bjorken variable. The inclusion of the N^3LO corrections reduces the
theoretical uncertainty of alpha_s determinations from non-singlet scaling
violations arising from the truncation of the perturbation series to less than
1%. We also study the predictions of the soft-gluon resummation, of
renormalization-scheme optimizations by the principle of minimal sensitivity
(PMS) and the effective charge (ECH) method, and of the Pade' summation for the
structure-function evolution kernels. The PMS, ECH and Pade' approaches are
found to facilitate a reliable estimate of the corrections beyond N^3LO.
