  We present the complete four-loop four-point amplitude in N=4
super-Yang-Mills theory, for a general gauge group and general D-dimensional
covariant kinematics, and including all non-planar contributions. We use the
method of maximal cuts --- an efficient application of the unitarity method ---
to construct the result in terms of 50 four-loop integrals. We give graphical
rules, valid in D-dimensions, for obtaining various non-planar contributions
from previously-determined terms. We examine the ultraviolet behavior of the
amplitude near D=11/2. The non-planar terms are as well-behaved in the
ultraviolet as the planar terms. However, in the color decomposition of the
three- and four-loop amplitude for an SU(N_c) gauge group, the coefficients of
the double-trace terms are better behaved in the ultraviolet than are the
single-trace terms. The results from this paper were an important step toward
obtaining the corresponding amplitude in N=8 supergravity, which confirmed the
existence of cancellations beyond those needed for ultraviolet finiteness at
four loops in four dimensions. Evaluation of the loop integrals near D=4 would
permit tests of recent conjectures and results concerning the infrared behavior
of four-dimensional massless gauge theory.
