Legged robots have the potential to cover terrain not accessible to wheel-based robots and vehicles. This makes them better suited to perform tasks, such as search and rescue, in real-world unstructured environments. Pneumatically-actuated, compliant robots are also more suited than their rigid counterparts to work in real-world unstructured environments with humans where unintentional contact may occur. This thesis seeks to combine the benefits of these two type of robots by implementing design methods to aid in the design choice of a 16 degree of freedom (DoF) compliant, continuum-joint quadruped. This work focuses on the design optimization, especially the definition of design metrics, for this type of robot. The work also includes the construction and closed-loop control of a four-DoF continuum-joint leg used to validate design methods.We define design metrics for legged robot metrics that evaluate their ability to traverse unstructured terrain, carry payloads, find stable footholds, and move in desired directions. These design metrics require a sampling of a legged-robot's complete configuration space. For high-DoF robots, such as the 16-DoF in evaluated in this work, the evaluation of these metrics become intractable with contemporary computing power. Therefore, we present methods that can be used to simplify and approximate these metrics. These approximations have been validated on a simulated four-DoF legged robot where they can tractably be compared against their full counterparts.Using the approximations of the defined metrics, we have performed a multi-objective design optimization to investigate the ten-dimensional design space of a 16-DoF compliant, continuum-joint quadruped. The design variables used include leg link geometry, robot base dimensions, and the leg mount angles. We have used an evolutionary algorithm as our optimization method which converged on a Pareto front of optimal designs. From these set of designs, we are able to identify the trade-offs and design differences between robots that perform well in each of the different design metrics. Because of our approximation of the metrics, we were able to perform this optimization on a supercomputer with 28 cores in less than 40 hours.We have constructed a 1.3 m long continuum-joint leg from one of the resulting quadruped designs of the optimization. We have implemented configuration estimation and control and force control on this leg to evaluate the leg payload capability. Using these controllers, we have conducted an experiment to compare the leg's ability to provide downward force in comparison with its theoretical payload capabilities. We then demonstrated how the torque model used in the calculation of payload capabilities can accurately calculate trends in force output from the leg.