  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.
