Recent advances in charm mixing and $CP$ violation at LHCb

After playing a pivotal role in the birth of the Standard Model in the 70's, the study of charm physics has undergone a revival during the last decade, triggered by a wealth of precision measurements from the charm and $B$ factories, and from the CDF and especially the LHCb experiments. In this article, we sum up how the unique phenomenology of charmed hadrons can be used to test the Standard Model and we review the latest measurements performed in this field by the LHCb experiment. These include the historic first observations of $CP$ violation and of a nonzero mass difference between the charmed neutral-meson mass eigenstates, the most precise determination of their decay-width difference to date, and a search for time-dependent $CP$ violation reaching the unprecedented precision of $10^{-4}$. These results challenge our comprehension of nonperturbative strong interactions, and their interpretation calls for further studies on both the theoretical and experimental sides. The upcoming upgrades of the LHCb experiment will play a leading role in this quest.