Initial state QED radiation aspects for future e +e − colliders

This white paper concerns theoretical and phenomenological aspects relevant to the physics of future e+e− colliders, in particular regarding initial-state QED radiation. The contributions each contain key technical aspects, and are formulated in a pedagogical manner so as to render them accessible also to those who are not directly working on these and immediately-related topics. This should help both experts and non-experts understand the theoretical challenges that we shall face at future e+e− colliders. Specifically, this paper contains descriptions of the treatment of initial state radiation from several Monte Carlo collaborations, as well as contributions that explain a number of more theoretical developments with promise of future phenomenological impact.