I present recent progress towards determining the planar S-matrix of maximally supersymmetric Yang-Mills theory, thanks to the rich interplay between its perturbative analytic properties in general kinematics, and its integrable structure in special kinematics. The former are related to cluster algebras, and allow for the computation of amplitudes with six/seven gluons up to six/four loops, whereas the latter yields all amplitudes in the multi-Regge limit at finite coupling.