This master's thesis provides a conclusive derivation of the dampened least-squares approach to inverse kinematics. Even though this approach is widely researched the underlying mathematics are not well covered. The dampened least-squares approach is proven to be very versatile. It gives the ability to express motion in spacial coordinates which renders motion creation much more feasible in contrast to motion generation in joint-space. Inverse kinematics can also be utilized as an abstraction to robot models and structures since motion in task space can be easily transferred to different robots. In addition to the derivation this thesis discusses extensions to the performance in terms of numerical stability, its uses for non-trivial robot structures (e.g., loopy robots) and the incorporation of nonlinear constraints. Furthermore, this thesis shows that the dampened least-squares inverse kinematics outperforms analytical approaches in terms of general usability.