  We present the three-loop remainder function, which describes the scattering
of six gluons in the maximally-helicity-violating configuration in planar N=4
super-Yang-Mills theory, as a function of the three dual conformal cross
ratios. The result can be expressed in terms of multiple Goncharov
polylogarithms. We also employ a more restricted class of "hexagon functions"
which have the correct branch cuts and certain other restrictions on their
symbols. We classify all the hexagon functions through transcendental weight
five, using the coproduct for their Hopf algebra iteratively, which amounts to
a set of first-order differential equations. The three-loop remainder function
is a particular weight-six hexagon function, whose symbol was determined
previously. The differential equations can be integrated numerically for
generic values of the cross ratios, or analytically in certain kinematics
limits, including the near-collinear and multi-Regge limits. These limits allow
us to impose constraints from the operator product expansion and multi-Regge
factorization directly at the function level, and thereby to fix uniquely a set
of Riemann-zeta-valued constants that could not be fixed at the level of the
symbol. The near-collinear limits agree precisely with recent predictions by
Basso, Sever and Vieira based on integrability. The multi-Regge limits agree
with the factorization formula of Fadin and Lipatov, and determine three
constants entering the impact factor at this order. We plot the three-loop
remainder function for various slices of the Euclidean region of positive cross
ratios, and compare it to the two-loop one. For large ranges of the cross
ratios, the ratio of the three-loop to the two-loop remainder function is
relatively constant, and close to -7.
