Stair-climbing is a difficult task for mobile robots to accomplish, particularly for legged robots. While quadruped robots have previously demonstrated the ability to climb stairs, none have so far been capable of climbing stairs of variable height while carrying all required sensors, controllers, and power sources on-board. The goal of this thesis was the development of a self-contained quadruped robot capable of detecting, classifying, and climbing stairs of any height within a specified range. The design process for this robot is described, including the development of the joint, leg, and body configuration, the design and selection of components, and both dynamic and finite element analyses performed to verify the design. A parameterized stair-climbing gait is then developed, which is adaptable to any stair height of known width and height. This behavior is then implemented on the previously discussed quadruped robot, which then demonstrates the capability to climb three different stair variations with no configuration change.