  This paper is part of an ongoing program to develop a theory of generalized
differential geometry. We consider the space $\mathcal{G}[X,Y]$ of Colombeau
generalized functions defined on a manifold $X$ and taking values in a manifold
$Y$. This space is essential in order to study concepts such as flows of
generalized vector fields or geodesics of generalized metrics. We introduce an
embedding of the space of continuous mappings $\mathcal{C}(X,Y)$ into
$\mathcal{G}[X,Y]$ and study the sheaf properties of $\mathcal{G}[X,Y]$.
Similar results are obtained for spaces of generalized vector bundle
homomorphisms. Based on these constructions we propose the definition of a
space $\mathcal{D}'[X,Y]$ of distributions on $X$ taking values in $Y$.
$\mathcal{D}'[X,Y]$ is realized as a quotient of a certain subspace of
$\mathcal{G}[X,Y]$.
