We present the first calculations of next-to-next-to-next-to-next-to-leading order (N4LO) contributions to anomalous dimensions of spin-N twist-2 operators in perturbative QCD. Specifically, we have obtained the respective non-singlet quark–quark anomalous dimensions at N=2 and N=3 to the fifth order in the strong coupling αs. These results set the scale for the N4LO contributions to the evolution of the non-singlet quark distributions of hadrons outside the small-x region, and facilitate a first approximate determination of the five-loop cusp anomalous dimension. While the N4LO coefficients are larger than expected from the lower-order results, their inclusion stabilizes the perturbative expansions for three or more light flavours at a sub-percent accuracy for αs<0.3.