Harnessing variable wind and solar resources for most of our energy use will require advanced energy storage options. Flow batteries have an attractive battery architecture due to their scalability, long cycle-life, and power-to-energy tunability. However, the low energy density of flow batteries (~10-50 Wh/kg practical) limits their application. A significant hurdle to high-energy density flow batteries is finding negolytes and posolytes that remain in the liquid state at extremely high concentrations of redox-active species. One strategy to overcome this hurdle is to employ eutectic mixing properties for the depression of melting points. This dissertation focuses on the prospects of using Na-K liquid metal alloy, which has a eutectic melting point of -13 °C, as the negolyte in a flow battery at room temperature. At room temperature, liquid Na-K has an extremely high capacity of 580 mAh/g and a reduction potential of -2.9 V vs. SHE, assuming K+ is the mobile ion. First, we discuss the stability and compatibility of K-β''-alumina with Na-K. We find K- β''-alumina to be stable in phase and morphology to K-β''-alumina and to be an overwhelming conductor of K+ vs. Na+ when in contact with Na-K. These findings allowed us to demonstrate the cycling of batteries using Na-K, K-β''-alumina, and model aqueous and nonaqueous posolytes. Second, we discuss the different components of area-specific resistance in a Na-K-based flow battery. The interfaces on both sides of the K-β''-alumina can be large sources of impedance. On the Na-K [verticle line] K-β''-alumina interface, although Na-K does not wet K-β''-alumina at room temperature, we find some form of reactive wetting that decreases the interfacial resistance significantly. On the posolyte [verticle line] K-β''-alumina interface, we find that if the posolyte is aqueous, ion exchange of hydrogen/hydronium species and K+ increases the interfacial resistance substantially over time, but this can be greatly decreased through modifying the solution chemistry. These findings allowed us to achieve a high power density for a battery of this kind, 65 mW cm^-2 at 22 °C and > 100 mW cm^-2 at 57 °C. Third, we discuss our efforts to find an ideal posolyte by employing eutectic mixing to decrease the melting points of benzoquinone derivatives, which are redox-active organic molecules. We find that a regular solution model that assumes immiscible solids has predictive power for finding eutectic melting temperatures and compositions for mixtures of multiple components.