  We construct the six dimensional Quantum Chromodynamics (QCD) Lagrangian in a
linear covariant gauge and subsequently renormalize it at two loops in the
MSbar scheme. The coupling constant corresponding to the gauge interaction is
asymptotically free for all numbers of quark fields, Nf. Analysing the
beta-functions yields a rich spectrum of fixed points. For instance, the
conformal window in the six dimensional theory is at Nf = 16 for the SU(3)
colour group. The critical theory structure is similar to that of an O(N)
scalar theory in eight dimensions. Using the large N expansion the latter is
shown to be in the same universality class as the Heisenberg ferromagnet.
Similarly using the large Nf expansion, six dimensional QCD is shown to be in
the same class as the two dimensional non-abelian Thirring model and four
dimensional QCD. Abelian gauge theories are also renormalized at high loops in
six and eight dimensions. It is shown that the gauge parameter only appears in
the electron anomalous dimension at one loop, similar to four dimensions.
