The performance of conventional restoration algorithms is highly degraded by the presence of outliers in the nominal noise process. A novel restoration approach that combines the properties of regularized and robust estimation schemes is introduced. Moreover, we address the potential of robust stabilizing functionals in preserving the detailed structure. Initially, the least-squares part of the regularized optimization scheme is modified according to the notion of M-estimation. An iterative algorithm is introduced for the derivation of the corresponding nonlinear estimate. The global convergence of this algorithm is rigorously studied. The robust regularized criterion provides nonlinear estimates, which do not suffer from artifacts due to the presence of outliers. In addition, the presence of the stabilizing functional enables the efficient suppression of such outliers. For the evaluation of the regularization parameter, several optimality measures are considered. Moreover, an adaptive structure for this parameter, which leads to high-quality estimates without significantly affecting the computational complexity or the global convergence of the algorithm, is introduced. To further account for the effective preservation of the detailed structure, the concept of M-estimators is embedded in both composite functionals involved in the regularized criterion. It is demonstrated that the corresponding error-penalizing strategy allows the reconstruction of sharp edges, while preserving the robust characteristics of the restoration algorithm. The capabilities of the robust regularized algorithms are demonstrated through restoration examples.