The search for new high-temperature alloys that can enable higher-efficiency/lower-emissions power generation has accelerated with the discovery of body-centered cubic (bcc) refractory High Entropy Alloys (HEAs). These many-component, non-dilute alloys in the Cr-Mo-W-V-Nb-Ta-Ti-Zr-Hf-Al family hold the potential for combining high strength and thermodynamic stability at high temperature with low density and room-temperature ductility, but searching the immense compositional space is daunting. Here, very recent theories and expanded thermodynamic tools are used to guide the discovery of new alloys satisfying the required suite of properties. The search and discovery method is first demonstrated for 5-component equicomposition alloys, identifying HfMoNbTaTi as the one alloy satisfying many constraints, with predicted properties agreeing with experiments. The design process then discovers new quinary and quarternary alloys in the Hf-Mo-Nb-Ta-Ti space having even better overall properties. One new quinary alloy is fabricated and shown to be single phase with high room temperature hardness, high melting point, and low density. More broadly, the new design process can further be used to explore millions of alloys with other desired multi-dimensional performance requirements. (C) 2022 The Author(s). Published by Elsevier Ltd on behalf of Acta Materialia Inc.