We develop a general framework for solving a variety of variational and computer vision problems involving framed space curves. Our approach is to study the global Riemannian and symplectic geometry of the moduli space of similarity classes of framed loops in Euclidean space. We show that this space is an infinite-dimensional Frechet manifold with a natural Kahler structure. The proof uses novel coordinates on the space of framed paths, which are used to locally identify the moduli space with the Grassmannian of 2-planes in an infinite-dimensional complex vector space. Results on the geometry of framed loop space are obtained, including a characterization of its sectional curvatures and an in-depth description of some natural Hamiltonian group actions on the space. We give a variety of applications of this structure, such as a classification of critical points of a generalization of the Kirchhoff elastic energy functional and a shape recognition algorithm for, e.g., protein backbones. We show connections between previous results by various authors on infinite-dimensional Kahler geometry, fluid dynamics and moduli spaces of linkages.