The anomalous magnetic moment of the muon ($a_{\mu}$), or muon g-2, is one of the most celebrated tests of the Standard Model (SM) of particle physics. I will discuss some recent developments in the long-standing discrepancy between experiment and the SM prediction of $a_{\mu}$. In particular, I will present a recent proposal for a new experiment to measure the running of the fine-structure constant in the space-like region via $\mu e \to mu e$ scattering which could provide an independent and competitive determination of the leading hadronic contribution to $a_{\mu}$. I will conclude arguing that the electron g-2 may be used to probe new physics as well, and discuss possible methods to improve the present limits on the g-2 of the tau lepton using radiative leptonic tau decays.
