  We compute the three-loop helicity amplitudes for the scattering of four
gluons in QCD. We employ projectors in the 't Hooft-Veltman scheme and
construct the amplitudes from a minimal set of physical building blocks, which
allows us to keep the computational complexity under control. We obtain
relatively compact results that can be expressed in terms of harmonic
polylogarithms. In addition, we consider the Regge limit of our amplitude and
extract the gluon Regge trajectory in full three-loop QCD. This is the last
missing ingredient required for studying single-Reggeon exchanges at
next-to-next-to-leading logarithmic accuracy.
