Status of the lepton $g-2$ and effects of hadronic corrections

The electron and muon anomalous magnetic moments (AMM) are measured in experiments and studied in the Standard Model (SM) with the highest precision accessible in particle physics. The comparison of the measured quantity with the SM prediction for the electron AMM provides the best determination of the fine structure constant. The muon AMM is more sensitive to the appearance of New Physics effects and, at present, there appears to be a three- to four-standard deviation between the SM and experiment. The lepton AMMs are pure relativistic quantum correction effects and therefore test the foundations of relativistic quantum field theory in general, and of quantum electrodynamics (QED) and SM in particular, with highest sensitivity. Special attention is paid to the studies of the hadronic contributions to the muon AMM which constitute the main source of theoretical uncertainties of the SM.