  Using cutting techniques we obtain the two-loop N=4 super-Yang-Mills helicity
amplitudes for four-gluon scattering in terms of scalar integral functions. The
N=4 amplitudes are considerably simpler than corresponding QCD amplitudes and
therefore provide a testing ground for exploring two-loop amplitudes. The
amplitudes are constructed directly in terms of gauge invariant quantities and
therefore remain relatively compact throughout the calculation. We also present
a conjecture for the leading color four-gluon amplitudes to all orders in the
perturbative expansion.
