In this thesis, we investigate the mobility challenges associated with robotic exploration of small solar system bodies, such as comets and asteroids. We open with a discussion on the surface environment of small bodies, and in particular, how their extremely weak gravity motivates hopping as a promising form of locomotion for long-distance traverses. We then propose an adaptable rover architecture called ``Hedgehog''---a minimalistic, internally-actuated, hopping rover designed for targeted mobility in such low-gravity environments. By applying internal torques to three mutually-orthogonal flywheels, the rover's chassis rotates, giving rise to ground reaction forces and various motion primitives, including long-range hopping, short precise tumbling, and small pose adjustments. We propose various models for analyzing the dynamics of these motion primitives and derive control laws for achieving desired motions. We then discuss various methods for conducting experiments in a reduced-gravity environment, including a custom six-degree-of-freedom laboratory test bed and parabolic flights. We validate our control laws in these test beds and demonstrate unprecedented motion accuracy for internally-actuated hoppers. Finally, we broaden our focus to general hopping platforms and consider various algorithmic tools for autonomous exploration. Specifically, we develop a suite of tools for motion planning, localization, and traversability analysis, with a careful attention on the various sources of model uncertainty and the complex dynamics of hopping trajectories. Despite the stochastic nature of bouncing dynamics, we demonstrate through high-fidelity simulations that a hopping rover can efficiently traverse highly irregular bodies that would otherwise be inaccessible to traditional rovers.