The reconstruction of discretized geometric shapes from empirical data, especially from image data, is important for many applications in medicine, biology, materials science, and other fields. During the last years, a number of techniques for performing such geometrical reconstructions and for conducting shape analysis have been developed. An important mathematical concept in this context are shape spaces. These are high-dimensional quotient manifolds with Riemannian structure, whose points represent geometrical shapes. Using suitable metrics and probability density functions on such manifolds, distances between shapes or statistical shape priors (for utilization in reconstruction tasks) can be defined. A frequently encountered situation is that instead of a set of discrete shapes a series of shapes is given, varying with some parameter (e.g. time). The corresponding mathematical object is a trajectory in shape space. For many analysis questions it is helpful to consider the shape trajectories as such (instead of individual shapes) - often together with co-varying parameters. The focus of this project is to develop new mathematical methods for the analysis, processing and reconstruction of empirically defined shape trajectories. By treating the trajectories as curves in shape space, we plan to exploit the rich geometric structure inherent to these spaces. In consequence, we expect the derived schemes to benefit from a compact encoding of constraints and a superior consistency as compared to their Euclidean counterparts.