  We develop a comprehensive geometric framework for defining spaces
$\mathcal{G}(M,E)$ of nonlinear generalized sections of vector bundles $E \to
M$ containing spaces of distributional sections $\mathcal{D}'(M, E)$. Our
theory incorporates classical differential geometric operations (like tensor
products, covariant derivatives and Lie derivatives), is localizable and fully
compatible with smooth and distributional tensor calculus. As an application to
the treatment of singular metrics, we calculate the curvature of the conical
metric used to describe cosmic strings.
