  Radiative corrections are essential for an accurate determination of $V_{ud}$
from superallowed $\beta$ decays. In view of recent progress in the
single-nucleon sector, the uncertainty is dominated by the theoretical
description of nucleus-dependent effects, limiting the precision that can
currently be achieved for $V_{ud}$. In this work, we provide a detailed account
of the electroweak corrections to superallowed $\beta$ decays in effective
field theory (EFT), including the power counting, potential and ultrasoft
contributions, and factorization in the decay rate. We present a first
numerical evaluation of the dominant corrections in light nuclei based on
Quantum Monte Carlo methods, confirming the expectations from the EFT power
counting. Finally, we discuss strategies how to extract from data the
low-energy constants that parameterize short-distance contributions and whose
values are not predicted by the EFT. Combined with advances in ab-initio
nuclear-structure calculations, this EFT framework allows one to systematically
address the dominant uncertainty in $V_{ud}$, as illustrated in detail for the
$^{14}$O $\to$ $^{14}$N transition.
