Since the experiments of Stern and Gerlach, magnetic moments of "elementary" particles have been important in our quest to understand subatomic physics. The first "QED loop calculation" was done by Schwinger to explain the larger than expected hydrogen hyperfine structure. The definitive measurement of the electron's anomalous magnetic moment by Kusch and Foley followed and agreed well with Schwinger's calculation of ae = (alpha / 2 pi). The muon's anomaly, which is sensitive to a broad range of physics beyond the standard model, will be discussed in a historical context, following the intellectual development of modern physics through the 20th century. Experiment E821 at Brookhaven will be described, as well as possible improvements, which are highly desirable since the present value of the muon anomaly, appears to differ from the standard-model value by 3.4 standard deviations.