In the search for physics beyond the Standard Model (SM), precision measurements play a key role. A small deviation in a rare or forbidden process, or generally in a transition which can be measured with great accuracy, may well be the first sign of a more fundamental theory, and potentially point to scales much higher than the ones directly within reach of current experiments. The proposed FCC-ee pushes this precision frontier a significant step further, with expected leaps in precision across flavour, electroweak, Higgs, and top physics. We review how these measurements will impact our understanding of beyond-SM physics, from the electroweak scale up to the grand unification scale.