Discovery of the Earth's Van Allen radiation belts by instruments flown on Explorer 1 in 1958 was the first major discovery of the Space Age. The observation of distinct inner and outer zones of trapped megaelectron volt (MeV) particles, primarily protons at low altitude and electrons at high altitude, led to early models for source and loss mechanisms including Cosmic Ray Albedo Neutron Decay for inner zone protons, radial diffusion for outer zone electrons and loss to the atmosphere due to pitch angle scattering. This scattering lowers the mirror altitude for particles in their bounce motion parallel to the Earth's magnetic field until they suffer collisional loss. A view of the belts as quasi‐static inner and outer zones of energetic particles with different sources was modified by observations made during the Solar Cycle 22 maximum in solar activity over 1989–1991. The dynamic variability of outer zone electrons was measured by the Combined Radiation Release and Effects Satellite launched in July 1990. This variability is caused by distinct types of heliospheric structure that vary with the solar cycle. The launch of the twin Van Allen Probes in August 2012 has provided much longer and more comprehensive measurements during the declining phase of Solar Cycle 24. Roughly half of moderate geomagnetic storms, determined by intensity of the ring current carried mostly by protons at hundreds of kiloelectron volts, produce an increase in trapped relativistic electron flux in the outer zone. Mechanisms for accelerating electrons of hundreds of electron volts stored in the tail region of the magnetosphere to MeVenergies in the trapping region are described in this review: prompt and diffusive radial transport and local acceleration driven by magnetospheric waves. Such waves also produce pitch angle scattering loss, as does outward radial transport, enhanced when the magnetosphere is compressed. While quasilinear simulations have been used to successfully reproduce many essential features of the radiation belt particle dynamics, nonlinear wave‐particle interactions are found to be potentially important for causing more rapid particle acceleration or precipitation. The findings on the fundamental physics of the Van Allen radiation belts potentially provide insights into understanding energetic particle dynamics at other magnetized planets in the solar system, exoplanets throughout the universe, and in astrophysical and laboratory plasmas. Computational radiation belt models have improved dramatically, particularly in the Van Allen Probes era, and assimilative forecasting of the state of the radiation belts has become more feasible. Moreover, machine learning techniques have been developed to specify and predict the state of the Van Allen radiation belts. Given the potential Space Weather impact of radiation belt variability on technological systems, these new radiation belt models are expected to play a critical role in our technological society in the future as much as meteorological models do today. Plain Language Summary Discovery of the Earth's Van Allen radiation belts by instruments flown on Explorer 1 in 1958 was the first major discovery of the Space Age. The dynamic properties of trapped outer zone electrons and the outer boundary of the inner zone proton population, along with source populations, have recently been studied in great detail by instruments on National Aeronautics and Space Administration's Van Allen Probes spacecraft, as well as other data sources like operational spacecraft designed for navigation and terrestrial weather forecasting. The vulnerability of the myriad of spacecraft that is strongly affected by space weather disruptions, as compared to 1958, has motivated the radiation belt community to develop essential improved models for forecasting the space environment we will inhabit in the 21st century and evaluate its impacts on our technological society. In this paper, we provide a review on historical background and recent advances in understanding and modeling acceleration, transport, and loss processes of energetic particles in the Earth's Van Allen radiation belts, followed by outstanding challenges for developing future radiation belt models. The findings on the fundamental physics of the Van Allen radiation belts potentially provide insights into understanding energetic particle dynamics at other magnetized planets in the solar system, exoplanets throughout the universe, as well as in astrophysical and laboratory plasmas. Given the potential Space Weather impact of radiation belt variability on technological systems, these new radiation belt models are expected to play a critical role in our technological society in the future much as meteorological models do today. Key Points A brief historical background on the discovery of the Van Allen radiation belts and their response to solar activity is introduced Recent advances in understanding mechanisms responsible for radiation belt electron acceleration, transport, and loss are reviewed Outstanding challenges for developing future radiation belt models are summarized