Grid-connected converters must remain coupled to the grid during a fault. They have to control the reactive currents injected to the grid because of its limited overload capacity. In particular, grid-forming (GFM) converters ride through the fault either preserving GFM operation or switching from GFM to grid-following (GFL) mode. In the GFM mode, the current cannot be controlled, and therefore, mechanisms as virtual impedances become necessary. This article proposes a converter with GFM and GFL capabilities that demonstrates effective voltage tracking in the GFM mode and current tracking in the GFL mode, good dynamic disturbance rejection response, and wide stability range and robustness in both modes. During a fault, the proposed converter switches from GFM to GFL mode in order to precisely control the injected currents. The current control is able to limit the current at the inception of severe faults. This characteristic is preserved for a wide range of grid impedances, thanks to its good robustness and wide stability margin. As a result, the controller can operate in a stand-alone mode, connected to a weak grid or connected to a stiff grid with a short-circuit ratio of up to 50. The fault ride-through capability of the converter is tested under severe symmetrical and asymmetrical faults.