ABSTRACT:   The Earth’s radiation belts were the first discovery of the space age with the launch of Explorer 1 in 1958. Later Explorers confirmed the discovery and led to an initial mapping of the region. Later missions, CRRES and SAMPEX, uncovered complexities and mysteries that are still being investigated. Discoveries and understandings achieved by the current Van Allen Probes science investigations are extensive. For a few of the discoveries, a mystery or controversy has essentially been resolved through a combination of new, high quality data and effective modeling of the processes. For one unanticipated behavior of the radiation belts, the third radiation belt, it is shown how modeling uncovered the explanation for how and why that occurred.  In addition, the long-standing issue of local electron acceleration can be explained using high-resolution measurements and modeling of local chorus waves. But there are many unresolved problems in the inner magnetosphere; a few of these unresolved problems are discussed and progress to date is shown. Although the Van Allen Probes are greatly enhancing our understanding of radiation belt processes, much remains to be done and there is a continuing need for high quality measurements that can complement the existing satellites.