The high atomic-scale complexity inherent in the aptly named complex concentrated alloys, or high entropy alloys, presents unique challenges in understanding (1) the structure and motion of defects that control mechanical properties and (2) the thermodynamic phase space encompassing stable, metastable, single, and multiphase alloys, possibly with chemical short range ordering. These factors plus the huge range of possible compositions makes computationally guided design of new high-performance alloys difficult but essential. Here, emerging concepts and theoretical frameworks for understanding defect structures, energies, and motion, and thermodynamics are discussed with a focus on yield strength and phase behavior. Pressing directions for future research are suggested to advance toward the predictive capabilities needed for alloy design.