Natural gamma radiation (NGR), one of the major geophysical parameters directly connected with cloud electrification and lightning initiation, is highly enhanced during thunderstorms. At low energies below 3 MeV, the enhancement of NGR is due to natural isotope radiation, and for energies up to 50 MeV, it is due to the operation of the newly discovered electron accelerators in the thunderclouds. For the first time, we present a comprehensive model of the enhanced fluxes of radiation incident on the earth's surface during thunderstorms. In addition to the already explained minute-long fluxes of high-energy electrons and gamma rays from relativistic runaway electron avalanches (RREA), we clarify also the origin of hour-long isotropic fluxes of low-energy gamma rays from the Rn-222 progenies. Also, as a direct evidence of RREA, we present photographs of optical emission during the development of electron-gamma ray cascades in the atmosphere. Natural radioactivity is a source of continuous exposure of human beings to radiation. Radiation protection of living organisms requires an understanding of all sources and possible ways of enhancement of the radiation levels that can double for several hours in the energy domain of hundreds of keV. Therefore individual irradiation doses can be exceeded during thunderstorms. The models used for the forecasting of thunderstorms and other severe atmospheric phenomena need an accurate account of the ionizing radiation in the atmosphere. The airglows can influence the operation of optical, fluorescence, and atmospheric Cherenkov telescopes and fluorescence detectors.