Human movement is a complex process influenced by physiological and psychological factors. The execution of movement is varied from person to person, and the number of possible strategies for completing a specific movement task is almost infinite. Different choices of strategies can be perceived by humans as having different degrees of quality, and the quality can be defined with regard to aesthetic, athletic, or health-related ratings. It is useful to measure and track the quality of a person's movements, for various applications, especially with the prevalence of low-cost and portable cameras and sensors today. Furthermore, based on such measurements, feedback systems can be designed for people to practice their movements towards certain goals. In this dissertation, I introduce symmetry as a family of measures for movement quality, and utilize recent advances in computer vision and differential geometry to model and analyze different types of symmetry in human movements. Movements are modeled as trajectories on different types of manifolds, according to the representations of movements from sensor data. The benefit of such a universal framework is that it can accommodate different existing and future features that describe human movements. The theory and tools developed in this dissertation will also be useful in other scientific areas to analyze symmetry from high-dimensional signals.