Quantitative susceptibility mapping (QSM) allows new insights into tissue composition and organization by assessing its magnetic property. Previous QSM studies have already demonstrated that magnetic susceptibility is highly sensitive to myelin density and fiber orientation as well as to para-and diamagnetic trace elements. Image resolution in QSM with current approaches is limited by the long acquisition time of 3D scans and the need for high signal to noise ratio (SNR) to solve the dipole inversion problem. We here propose a new total-generalized-variation (TGV) based method for QSM reconstruction, which incorporates individual steps of phase unwrapping, background field removal and dipole inversion in a single iteration, thus yielding a robust solution to the reconstruction problem. This approach has beneficial characteristics for low SNR data, allowing for phase data to be rapidly acquired with a 3D echo planar imaging (EPI) sequence. The proposed method was evaluated with a numerical phantom and in vivo at 3 and 7 T. Compared to total variation (TV), TGV-QSM enforced higher order smoothness which yielded solutions closer to the ground truth and prevented stair-casing artifacts. The acquisition time for images with 1 mm isotropic resolution and whole brain coverage was 10 s on a clinical 3 Tesla scanner. In conclusion, 3D EPI acquisition combined with single-step TGV reconstruction yields reliable QSM images of the entire brain with 1 mm isotropic resolution in seconds. The short acquisition time combined with the robust reconstruction may enable new QSM applications in less compliant populations, clinical susceptibility tensor imaging, and functional resting state examinations. (C) 2015 Elsevier Inc. All rights reserved.