  Nuclei will play a prominent role in searches for physics beyond the Standard
Model as the active material in experiments. In order to reliably interpret new
physics signals, one needs an accurate model of the underlying nuclear
dynamics. In this review, we discuss recent progress made with quantum Monte
Carlo approaches for calculating the electroweak structure of light nuclei. We
place particular emphasis on recent $\beta$-decay, muon capture,
$0\nu\beta\beta$-decay, and electron scattering results.
