The recent close encounters of Pioneer-10 (December 1973) and Pioneer-11 (December 1974) with the planet Jupiter provided the first in situ observations of zenomagnetically trapped particle radiation. Such observations represented a major advance in planetary research. Prior estimates of radiation intensities (particle fluxes) at Jupiter had necessarily relied (in the case of electrons) upon inferences from Jovian decimetric radio emission observed at the Earth and (in the case of protons) upon postulates for the numerical scaling from terrestrial proton intensities. The Pioneer-10 and Pioneer-11 observations have stimulated continuing theoretical efforts to understand the reported findings and to extrapolate from them to other planets and other epochs. While the analysis of trapped-radiation data from the Pioneer spacecraft is far from being completed, a consensus has developed with respect to the physical mechanisms that must be considered. The observed radiation belts seem to be populated by radial diffusion from an external source. The diffusion coefficient seems to be that derived from fluctuations in the polarization electric field produced by neutral winds in the Jovian ionosphere, which is coupled to the magnetosphere by equipotential B-field lines. Radiation-belt electrons lose energy and change their equatorial pitch angles by virtue of synchrotron emission. Radiation-belt ions and electrons both may be subject to pitch-angle diffusion caused by waves that the respective particle anisotropies have created through plasma instabilities. Finally, radiation-belt ions and electrons seem to experience absorption by the inner Jovian satellites (moons) in a manner that may depend upon the species and energy of the incident radiation-belt particle. It is not yet known whether satellite-associated clouds of sodium and sulfur contribute substantially to the inferred particle absorption. Also still open is the question of whether the satellites provide a substantial source of radiation-belt particles. Moreover, there remains doubt concerning the configuration of the outer Jovian magnetosphere and the influence of this configuration on the zenomagnetic trapping of energetic charged particles.