Nyquist ghosts are a longstanding problem in a variety of fast MRI experiments that use echo-planar imaging (EPI). Recently, several structured low-rank matrix modeling approaches have been proposed that achieve state-of-the-art ghost-elimination, although the performance of these approaches is still inadequate in several important scenarios. We present a new structured low-rank matrix recovery ghost correction method that we call Robust Autocalibrated LORAKS (RAC-LORAKS). RAC-LORAKS incorporates constraints from autocalibration data to avoid ill-posedness, but allows adaptation of these constraints to gain robustness against possible autocalibration imperfections. RAC-LORAKS is tested in two challenging scenarios: highly-undersampled multi-channel EPI of the brain, and cardiac EPI with a double-oblique slice orientation. Results show that RAC-LORAKS can provide substantial improvements over existing ghost correction methods, and potentially enables new imaging applications that were previously confounded by ghost artifacts.