  We present the O(alphas^4) virtual QCD corrections to the scattering process
of massless quark qqbar -> gg due to the interference of tree and two-loop
amplitudes and to the self-interference of one-loop amplitudes. We work in
conventional dimensional regularisation and our results are renormalised in the
MSbar scheme. The structure of the infrared divergences agrees with that
predicted by Catani while expressions for the finite remainder are given for
the qqbar -> gg and the qg -> qg (gqbar -> gqbar) scattering processes in terms
of logarithms and polylogarithms that are real in the physical region. These
results, together with those previously obtained for quark-quark scattering,
are important ingredients in the next-to-next-to-leading order contribution to
inclusive jet production at hadron colliders.
