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Herein, we critically evaluate computational and experimental studies in the emerging field of high-entropy ultra-high-temperature ceramics. High-entropy ultra-high-temperature ceramics are candidates for use in extreme environments that include temperatures over 2,000°C, heat fluxes of hundreds of watts per square centimeter, or irradiation from neutrons with energies of several megaelectron volts. Computational studies have been used to predict the ability to synthesize stable high-entropy materials as well as the resulting properties but face challenges such as the number and complexity of unique bonding environments that are possible for these compositionally complex compounds. Experimental studies have synthesized and densified a large number of different high-entropy borides and carbides, but no systematic studies of composition-structure-property relationships have been completed. Overall, this emerging field presents a number of exciting research challenges and numerous opportunities for future studies.