A review of the experimental and theoretical determinations of the anomalous magnetic moment of the muon is given. The anomaly is defined by a = (g − 2)/2, where the Landé g-factor is the proportionality constant that relates the spin to the magnetic moment. For the muon, as well as for the electron and tauon, the anomaly a differs slightly from zero (of the order 10⁻³) because of radiative corrections. In the Standard Model, contributions to the anomaly come from virtual 'loops' containing photons and the known massive particles. The relative contribution from heavy particles scales as the square of the lepton mass over the heavy mass, leading to small differences in the anomaly for e, μ and τ. If there are heavy new particles outside the Standard Model which couple to photons and/or leptons, the relative effect on the muon anomaly will be ~ (m_μ/m_e)² ≈ 43 × 10³ larger compared with the electron anomaly. Because both the theoretical and experimental values of the muon anomaly are determined to high precision, it is an excellent place to search for the effects of new physics or to constrain speculative extensions to the Standard Model. Details of the current theoretical evaluation and of the series of experiments that culminates with E821 at the Brookhaven National Laboratory, are given. At present the theoretical and the experimental values are known with a similar relative precision of 0.5 ppm. There is, however, a 3.4 standard-deviation difference between the two, strongly suggesting the need for continued experimental and theoretical study.