Extraterrestrial radiation is attenuated by the atmosphere in different proportions depending mainly on the solar zenith angle and the altitude of the measurement point. In this work, very high and extreme total solar irradiance measurements are presented that, on some days, surpassed the Solar Constant corrected by the actual Sun-Earth distance (CSC). This paper reports a detailed analysis of ground-based measurements of cloud-enhanced solar global irradiation in NE Brazil in the cities of Agua Branca, Santana do Ipanema, Palmeira dos Indios, Laje, Pao de Acucar, Arapiraca, Coruripe, and Maceio from January to December 2008. Measurements were made at 1-mm intervals using Eppley and B & W pyranometers with an estimated uncertainty of 5%. It was found that a) the phenomenon is not uncommon and that it occurs on at least one-third of the days in a given month; b) the cumulative duration (number of I-min consecutive events) can reach 34 min; d) there is a clear seasonal effect, and the probability of occurrence on a monthly basis shows two peaks, one in April and another in October; e) the most extreme solar radiation was 1650 W/m(2) in Agua Branca, which is approximately 350 W/m(2) above the extraterrestrial solar irradiation; and f) a strong asymmetry was detected in the occurrence profile between the morning and afternoon. Finally, the observation of extreme values should be taken into account in the study of solar radiation effects related to the UV index and biological effects, among others. Extreme radiation above the extraterrestrial irradiance lasting up to 30 min can mean very high and potentially dangerous UV (UVI) indexes, even in situations with seemingly adequate cloudiness. (c) 2015 Elsevier Ltd. All rights reserved.