Theoretical and experimental status of rare charm decays

Rare charm decays offer the unique possibility to explore flavour-changing neutral-currents in the up-sector within the Standard Model and beyond. Due to the lack of effective methods to reliably describe its low energy dynamics, rare charm decays have been considered as less promising for long. However, this lack does not exclude the possibility to perform promising searches for New Physics \textit{per se}, but a different philosophy of work is required. Exact or approximate symmetries of the Standard Model allow to construct clean null-test observables, yielding an excellent road to the discovery of New Physics, complementing the existing studies in the down-sector. In this review, we summarise the theoretical and experimental status of rare charm $|\Delta c|=|\Delta u|=1$ transitions, as well as opportunities for current and future experiments such as LHCb, Belle II, BES III and the FCC-ee. We also use the most recent experimental results to report updated limits on lepton-flavour conserving and lepton-flavour violating Wilson coefficients.