We present a Riemannian framework for geometric shape analysis of curves, functions, and trajectories on nonlinear manifolds. Since scalar functions and trajectories can also have important geometric features, we use shape as an all-encompassing term for the descriptors of curves, scalar functions and trajectories. Our framework relies on functional representation and analysis of curves and scalar functions, by square-root velocity fields (SRVF) under the Fisher–Rao metric, and of trajectories by transported square-root vector fields (TSRVF). SRVFs are general functional representations that jointly capture both the shape (geometry) and the reparameterization (sampling speed) of curves, whereas TSRVFs also capture temporal reparameterizations of time-indexed shapes. The space of SRVFs for shapes of curves becomes a subset of a spherical Riemannian manifold under certain special constraints. A fundamental tool in shape analysis is the construction and implementation of geodesic paths between shapes. This is used to accomplish a variety of tasks, including the definition of a metric to compare shapes, the computation of intrinsic statistics for a set of shapes, and the definition of probability models on shape spaces. We demonstrate our approach using several applications from computer vision and medical imaging including the analysis of (1) curves, (2) human growth, (3) bird migration patterns, and (4) human actions from video surveillance images and skeletons from depth images.