The ongoing transition of robotic systems from structured industrial settings to unstructured human-centric environments necessitates novel approaches to ensure adaptability, safety, and robustness. This thesis supports the view that geometrically coherent representations are pivotal in addressing these challenges. Therefore, this work presents a unified geometric framework for robot manipulation using Conformal Geometric Algebra (CGA), a mathematical language that seamlessly integrates geometric primitives (points, lines, planes, spheres) and rigid-body transformations (rotations, translations, scaling) into a single algebraic structure. By leveraging CGA's capacity to encode geometric intuition directly into algebraic operations, we address key challenges in robotics, including kinematics, dynamics, optimal control, and cooperative manipulation. Our results enable intuitive modeling of manipulation tasks through geometric primitives and their similarity transformations, which unify scaling, rotation, and translation. The presented modeling approaches generalize from single-, to dual-, and multi-arm systems. We show how this approach can be used in optimal control formulations as well as force control applications, where task objectives are formulated uniformly across primitives, avoiding case-specific adjustments and enabling compliant interactions. Practical contributions include the gafro library, an open-source CGA implementation tailored for robotics. The contributions not only provide theoretical advancements in geometric robotics but also deliver practical tools for real-world deployment. We demonstrate this by solving tasks around dual-arm admittance control and tactile ergodic surface coverage.