Scattering amplitudes in the high-energy limit can be described in terms of their singularity structure in the complex angular momentum plane, consisting of Regge poles and cuts. In QCD, gluon Reggeization has long been understood as a manifestation of a Regge pole, but until recently Reggeization violation remained largely obscure. New methods, based on iterative solution of rapidity evolution equations, facilitate direct computation of components of the amplitude which are mediated by multi-Reggeon exchange, a manifestation of Regge cuts. Upon disentangling the Regge cut from the pole we are now able to extract the pole parameters from state-of-the-art fixed- order computations (3 loops) and make predictions regarding certain components of the amplitude to higher loop orders. In this talk I review the key ideas which led to this progress, describe where we stand in exploring the structure of 2 → 2 and 2 → 3 amplitudes in the (multi-) Regge limit, and comment on the interplay between this research and the study of infrared factorization.