I will describe the geometry and topology of a general class of U-folds compatible with the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. We have named these U-folds as ``geometric U-folds''. The topology of geometric U-folds depends on certain fiber bundles which twist the local string theory fields and which we fully characterize. Smooth non-trivial U-folds of this type can exist only in theories where both the scalar and the space-time manifolds have non-trivial fundamental group and the scalar map is homotopically non-trivial. Requiring consistency with global Supergravity implies that geometric U-folds must be globally glued by using discrete subgrups of the continuous U-duality group, in agreement with String Theory requirements. I will also discuss the pre-moduli space of geometric U-folds, which can be identified with certain twists of character varieties that have not been studied in the literature. Finally, I will briefly discuss a generalization of geometric U-folds beyond smooth manifolds in terms of locally constant sheaves and their twisted cohomology on local orbifolds.