Measurements of the electron magnetic moment (the "g-value") probe the electron’s interaction with the fluctuating quantum vacuum. With a quantum electrodynamics calculation, they provide the most accurate determination of the fine structure constant. Comparisons with independent determinations of the fine structure constant are among the most precise tests of any physical theory. This talk will present an experiment that measures the g-value with an accuracy of 0.3 parts per trillion. The experiment involves trapping a single electron, resolving quantum jumps between harmonic oscillator eigenstates, and evaluating systematic effects related to the coupling of the electron’s motion to a microwave cavity. I will conclude with future precision measurement goals including promising new techniques.