  We compute the next-to-next-to-leading order (NNLO) contributions to the
three splitting functions governing the evolution of unpolarized non-singlet
combinations of quark densities in perturbative QCD. Our results agree with all
partial results available in the literature. We find that the correct leading
logarithmic (LL) predictions for small momentum fractions x do not provide a
good estimate of the respective complete results. A new, unpredicted LL
contribution is found for the colour factor d^{abc}d_{abc} entering at three
loops for the first time. We investigate the size of the corrections and the
stability of the NNLO evolution under variation of the renormalization scale.
Except for very small x the corrections are found to be rather small even for
large values of the strong coupling constant, in principle facilitating a
perturbative evolution into the sub-GeV regime.
