At the heart of my research lies the study of Riemannian geometries on infinite-dimensional manifolds. Exhibiting these Riemannian structures can shed new light on many important topics. My research in falls into two broad categories. a. Shape and Functional Data Analysis. The goal of shape analysis is to develop methods to classify, compare and describe geometric shapes. The space of all shapes of some certain type is usually an infinite dimensional manifold. Thus Riemannian geometry can provide a powerful tools to handle the tasks and problems that arise in this area. b. Euler Equations. Many prominent partial differential equations -- including Euler's equation for the motion of an incompressible fluid -- admit a variational formulation as geodesic equations on an infinite-dimensional Riemannian manifold. In my research I investigate relations between analytical properties of such PDEs and geometric properties of the underlying Riemannian spaces. In particular I study local and global well-posedness properties of these equations, the corresponding geodesic distances and Fredholm properties of the Riemannian exponential mapping.