  The precise knowledge of the QED initial state corrections is of instrumental
importance in studying high luminosity measurements in $e^+e^-$ annihilation at
facilities like LEP, the International Linear Collider ILC, CLIC, a Giga-$Z$
facility, and the planned FCC\_ee. Logarithmic corrections of up to $O(\alpha^6
L^5)$ are necessary with various subleading terms taken into account. This
applies to both the inclusive measurement of processes like $e^+ e^-
\rightarrow \gamma^*/Z^*$ and also the forward-backward asymmetry. As has been
shown recently, techniques from massive QCD, such as the computation of massive
on-shell operator matrix elements, can be used for these calculations. We give
an introduction to this topic and present both the calculation methods and the
numerical corrections having been reached so far.
