The muon anomalous magnetic moment is one of the most precisely measured quantities in particle physics. Recent high precision measurements (0.54ppm) at Brookhaven reveal a ``discrepancy'' by 3 standard deviations from the electroweak Standard Model which could be a hint for an unknown contribution from physics beyond the Standard Model. This triggered numerous speculations about the possible origin of the ``missing piece''. The remarkable 14-fold improvement of the previous CERN experiment, actually animated a multitude of new theoretical efforts which lead to a substantial improvement of the prediction of a_mu. The dominating uncertainty of the prediction, caused by strong interaction effects, could be reduced substantially, due to new hadronic cross section measurements in electron-positron annihilation at low energies. After an introduction and a brief description of the principle of the experiment, I present a major update and review the status of the theoretical prediction and discuss the role of the hadronic vacuum polarization effects and the hadronic light--by--light scattering contribution. Prospects for the future will be briefly discussed. As, in electroweak precision physics, the muon g-2 shows the largest established deviation between theory and experiment at present, it will remain one of the hot topics for further investigations.