This paper describes the development of a flying and walking robot, called the dynamic underactuated flying-walking (DUCK) robot. The DUCK robot combines a high-mobility flying platform, such as a quadcopter (quadrotor helicopter), with passive-dynamic legs to create a versatile system that can fly and walk. One of the advantages of passive-dynamic legs for walking is that additional actuators are not needed for terrestrial locomotion. Herein, a mathematical model is presented and simulations are used to help design a prototype robot. Experimental results demonstrate the feasibility of combining an aerial platform with passive-dynamic legs to create an effective flying and walking robot. In particular, two modes of walking are demonstrated: (1) passive walking down inclined surfaces for low-energy terrestrial locomotion, and (2) active (powered) walking by leveraging the capabilities of the flying platform, where thrust from the quadcopter's rotors enables the DUCK robot to take steps and walk on flat surfaces or up inclined surfaces.