How can we make a robot that can go anywhere on its own? This thesis presents several new behaviors on the RHex robot that greatly increase the variety of obstacles that it can overcome, including vertical jumps, flips, leaps onto and across ledges, aerial reorientations, and proprioceptively-aware behaviors. These behaviors inspire new tools to model and understand their transitional nature, wherein it is no longer useful to think of each step as being an equal part of a steady state gait. Legged robots will necessarily experience a variety of changing contact conditions as they locomote in complex environments epitomized by the rocky, sandy desert. Drawing on the much more mature literature of robot manipulation, this thesis presents the new modeling paradigm of "self-manipulation" that formally generates analytical equations of motion across all contact states. The framework is amenable to many ubiquitous simplifying assumptions (such as rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs) to reduce the complexity of these models despite the obvious physical inaccuracies that each incurs. Nevertheless the models capture enough of the physical world to represent the challenges confronting interesting behaviors in a qualitatively correct manor, including the effects of impulsive transitions between the various contact modes. More than numerical simulation, our goal is the distillation of these physically parametrized models into formal design insights (platform design, behavior design, and controller design), utilizing a variety of analytical and numerical methods. These behaviors are only possible with a robot designed to be both robust and powerful, and they make use of the unique capability of legged machines to interact with the environment in varied and, possibly, unpredictable ways. Careful actuator modeling is needed to achieve such acrobatic results, and so this thesis presents a spectrum of motor sizing tasks to ensure that the platform is up to the task. These tools are used to gain insight into various dynamic transitions for RHex, and we conjecture that their generalization will be of importance for a broad class of legged robots operating in remote and unstructured terrain.