The SAGEX review on scattering amplitudes Chapter 15: The multi-Regge limit

We review the Regge and multi-Regge limit of scattering amplitudes in gauge theory, focusing on QCD and its maximally supersymmetric cousin, planar N=4 super-Yang-Mills theory. We identify the large logarithms that aredeveloped in these limits, and the progress that has been made in resummingthem, towards next-to-next-to-leading logarithms for BFKL evolution in QCD,as well as all-orders proposals in planarN=4 super-Yang-Mills theory andthe perturbative checks of those proposals. We also cover the application ofsingle-valued multiple polylogarithms to this important kinematical limit ofparticle scattering.