Running with compliant curved legs involves the progression of the center of pressure, the changes of both the leg's stiffness and effective rest length, and the shift of the location of the maximum stress point along the leg. These phenomena are product of the geometric and material properties of these legs, and the rolling motion produced during stance. We examine these aspects with several reduced-order dynamical models to relate the leg's design parameters (such as normalized foot radius, leg's effective stiffness, location of the maximum stress point and leg shape) to running performance (such as robustness and efficiency). By using these models, we show that running with compliant curved legs can be more efficient, robust with fast recovery behavior from perturbations than running with compliant straight legs. Moreover, the running performance can be further improved by tuning these design parameters in the context of running with rolling. The results shown in this work may serve as potential guidance for future compliant curved leg designs that may further improve the running performance.