Most robotic manipulators, and especially those designed with autonomous operation in mind, consist of a series of joints that rotate about a single axis, also known as revolute joints. These mechanisms give robotic manipulators the degrees of freedom and versatility similar to that of the human arm, which they are designed to outperform. However, this results in a geometry of motion or kinematics that makes all aspects of robotic manipulation challenging from a computational perspective. A major part of this challenge lies in the fact that computing joint configurations adhering to a specific set of constraints (i.e., gripper pose) is a non-trivial problem. The procedure of finding feasible joint configurations and the mathematical problem associated with it are known as inverse kinematics - a core part of motion planning, trajectory optimization, calibration and other important challenges in successfully performing robotic manipulation. In recent years, the overall decrease of computation time required to perceive and process environmental and proprioceptive information has helped realize the potential of robotic manipulation in dynamic environments. Concurrently, a new standard in manipulator design has emerged, where additional degrees of freedom are added in order to increase their overall dexterity and capacity for motion. These two developments have vastly increased the requirements for inverse kinematics algorithms, which are now expected to deal with infinite solution spaces and difficult, nonlinear constraints. On the other hand, the addition of degrees of freedom in recent robot designs has enabled algorithms to search for locally optimal configurations with respect to some performance criteria in an infinitely large solution space. This property has motivated approaches that leverage non-Euclidean geometries to replace conventional constraints and optimization criteria, thereby overcoming computational bottlenecks and common failure modes. The contributions presented in this thesis propose three such approaches, that aim to develop new ways of looking at the problems associated with inverse kinematics through the use of geometric representations that are not widely utilized in robotic manipulation.