  We compute the next-to-next-to-leading order (NNLO) contributions to the
splitting functions governing the evolution of the unpolarized flavour-singlet
parton densities in perturbative QCD. The exact expressions are presented in
both Mellin-N and Bjorken-x space. We also provide accurate parametrizations
for practical applications. Our results agree with all partial results
available in the literature. As in the non-singlet case, the correct leading
logarithmic predictions for small momentum fractions x do not provide good
estimates of the respective complete splitting functions. We investigate the
size of the corrections and the stability of the NNLO evolution under variation
of the renormalization scale. The perturbative expansion appears to converge
rapidly at x >~ 10^-3. Relatively large third-order corrections are found at
smaller values of x.
