The lack of periodicity and long-range order poses significant challenges in explaining and modeling the properties of metallic glasses. Conventional modeling of nonexponential relaxation with stretched exponents leads to inconsistencies and rarely offers information on microscopic properties. Instead, using quasi-static anelastic relaxation, we have obtained relaxation-time spectra over >10 orders of magnitude of time for several metallic glasses. The spectra enable us to examine in microscopic detail the distribution of shear transformation zones and their properties. They reveal an atomically-quantized hierarchy of shear transformation zones, providing insights into the effect of structural relaxation and rejuvenation, the origin of plasticity and the mechanisms of the alpha and beta relaxation.