Robots are complex, high-dimensional systems governed by nonlinear, underactuated dynamics evolving on non-Euclidean manifolds, posing numerous challenges for control synthesis and analysis. While optimization-based methods of control can flexibly accommodate diverse dynamics, costs, and constraints, they often demand coarse approximations or powerful onboard processors due to their relatively poor computational effciency. Meanwhile, learning-based controllers require offline training that is often brittle and computationally burdensome. Conversely, explicit, analytical control laws with negligible computational overhead often perform robustly, but they are typically only applicable to individual systems (or a narrow class), limiting their broader usefulness. However, robots are not black-box nonlinear control systems—rather, their dynamics enjoy powerful properties (e.g., symmetry and mechanical structure) that can provide traction on control design problems. In this thesis, we explore the role of geometric methods in mitigating many of the above drawbacks, across both analytical and data-driven methods. We study the role of symmetry in systematically identifying effective abstractions for trajectory planning (“flat outputs”) in underactuated mechanical systems and explore applications to aerial manipulation. We also synthesize explicit tracking controllers for mechanical systems evolving on homogeneous Riemannian manifolds, and certify the almost global asymptotic stability of cascades (commonly seen in the closed-loop dynamics of hierarchical controllers). Lastly, we accelerate the training of tracking controllers via reinforcement learning using symmetry reduction, also improving the converged policy. In each method, a geometric perspective enables us to explainably construct abstractions that reduce dimensionality, enforce structure, and capture essential features, ultimately representing the system or problem in a form more convenient for analysis or design. Such reduced representations typically improve computational efficiency, while also encouraging generality over a broader class of systems and affording insight into why prior handcrafted approaches succeeded. Such realizations can also guide mechanical design, closing the control-morphology feedback loop and leading to synergies between a robot’s embodiment and its controller. By combining explainable abstractions with scalable computation, such methods build towards a future in which robotic systems move through their surroundings as capably and dynamically as their counterparts in Nature.