Random body-centered-cubic (BCC) "High Entropy" alloys are a new class of alloys, some having high strength and good ductility at room temperature and some having exceptional high-temperature strength. There are no theories of strengthening of screw dislocations in BCC metals that span naturally from the dilute limit to the multi-component, non-dilute concentrations typical of the high-entropy domain. Here, such a theory is developed and validated. Unlike low-temperature elemental BCC metals and very dilute BCC alloys, strength is not controlled by the kink-pair nucleation mechanism. Rather, screw dislocations naturally adopt a kinked structure as the minimum total energy configuration in the field of random alloying atoms. The characteristic length and energy scales for the low-energy kinked screw dislocation are derived for random alloys, leading to a characteristic spacing of both kinks and cross-kinks that depends on the kink formation energy and a characteristic collective solute/screw dislocation interaction energy parameter. Glide motion of this initially-kinked screw dislocation occurs via Peierls-type motion, lateral kink glide, and failure of cross-kinks. All these features are observed in molecular dynamics simulations. The resulting strength versus temperature, strain rate, and composition is analytic. The theory is validated by comparison to experiments on non-dilute Fe-Si, Nb-Mo, Nb-W, and Ti-Nb-Zr-based high entropy alloys versus composition and temperature. The theory provides a framework for tailored design of new high-performance BCC alloys. (C) 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.