The body centered cubic (BCC) high entropy alloys (HEAs) MoNbTaW and MoNbTaVW show exceptional strength retention up to 1900K. The mechanistic origin of the retained strength is unknown yet is crucial for finding the best alloys across the immense space of BCC HEA compositions. Experiments on Nb-Mo, Fe-Si and Ti-Zr-Nb alloys report decreased mobility of edge dislocations, motivating a theory of strengthening of edge dislocations in BCC alloys. The strength of BCC HEAs can be controlled by edge dislocations, especially at high temperatures, because the random field of solutes in these high-concentration alloys creates large energy barriers for thermally-activated edge glide. A parameter-free theory for this mechanism of edge motion in BCC alloys qualitatively and quantitatively captures the strength versus temperature for both MoNbTaW and MoNbTaVW alloys and other BCC HEAs. A reduced analytic version of the theory, embodying the edge-strengthening mechanism, then enables screening over > 600,000 compositions in the Mo-Nb-Ta-V-W family to identify promising new compositions with high retained strength and/or reduced mass density. Overall, the theory reveals an unexpected mechanism - edge dislocation motion in the random alloy - as responsible for high temperature strength in BCC alloys and paves the way for theory-guided design of new high-performance HEAs. (C) 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.