  We present the details of the analytic calculation of the three-loop
angle-dependent cusp anomalous dimension in QCD and its supersymmetric
extensions, including the maximally supersymmetric $\mathcal{N}=4$ super
Yang-Mills theory. The three-loop result in the latter theory is new and
confirms a conjecture made in our previous paper. We study various physical
limits of the cusp anomalous dimension and discuss its relation to the
quark-antiquark potential including the effects of broken conformal symmetry in
QCD. We find that the cusp anomalous dimension viewed as a function of the cusp
angle and the new effective coupling given by light-like cusp anomalous
dimension reveals a remarkable universality property -- it takes the same form
in QCD and its supersymmetric extensions, to three loops at least. We exploit
this universality property and make use of the known result for the three-loop
quark-antiquark potential to predict the special class of nonplanar corrections
to the cusp anomalous dimensions at four loops. Finally, we also discuss in
detail the computation of all necessary Wilson line integrals up to three loops
using the method of leading singularities and differential equations.
