In this paper, a high-resolution method for solving potentially ill-posed inverse problems is proposed. This method named FO-D-MUSIC allows for localization of brain current sources with unconstrained orientations from surface electroencephalographic (EEG) or magnetoencephalographic (MEG) data using spherical or realistic head geometries. The FO-D-MUSIC method is based on the following: 1) the separability of the data transfer matrix as a function of location and orientation parameters, 2) the fourth-order (FO) virtual array theory, and 3) the deflation concept extended to FO statistics accounting for the presence of potentially but not completely statistically dependent sources. Computer results display the superiority of the FO-D-MUSIC approach in different situations (very closed sources, small number of electrodes, additive Gaussian noise with unknown spatial covariance, etc.) compared to classical algorithms.