Coupling of light from free space to optical fibers is essential for many applications, while commonly used step-index optical fibers provide insufficient coupling efficiencies especially at large angles of incidence. Here, we demonstrate record-high coupling efficiencies achieved with dielectric nanostructures located on single-mode fiber end faces. We introduce a novel approach that allows fabricating dielectric nanostructures at the facet of a step-index optical fiber via an extended version of planar electron-beam based lithography. We demonstrate polarization- and angle-independent coupling of light into the fiber across a wide range of angles as large as 80°. We support our experimental results with an analytical model and extensive numerical simulations. Our results reveal the key properties of nanostructure-empowered fibers that may improve the performance of many optical devices requiring efficient collection of light, including quantum technologies (single-photon collection) or biophotonics (in vivo imaging). Our approach can be extended to other materials and geometries, merging fiber optics with high-index dielectric metasurfaces, allowing for unprecedented functionalities for the efficient control of light.