Charged Lepton Flavour Conservation and Lepton Flavour Universality are observed symmetries of the Standard Model (SM) of particle physics, with no underlying theoretical principle enforcing them. Lepton Flavour Violation (LFV) is observed in the neutrino sector, through the spontaneous oscillation of neutrinos between the three flavour families. In addition, a series of measurements by $B$ factories challenged the universality of lepton couplings to the weak interaction, sparking interest about possible Lepton Flavour Universality Violation (LFUV). A thorough scrutiny of these symmetries in the charged lepton sector represents a stringent test of the SM. Any confirmed deviation could be interpreted as the first manifestation of New Physics.

In this review, results on LFV and LFUV from the ATLAS and CMS collaborations, published between September 2023 and June 2024 are presented. Two searches for the LFV couplings $e\mu qt$ and $\mu\tau qt$ in the Effective Field Theory framework by the CMS, respectively ATLAS collaborations, as well as an updated search for the LFV $\tau\rightarrow 3\mu$ decay by CMS are reported. The single most precise test of lepton flavour universality in the decays of $W^\pm$ bosons to electrons and muons by the ATLAS collaboration is reviewed. Finally, two tests of LFUV with the first $R(J/\psi)$ and $R(K)$ measurements by the CMS collaboration are presented.