Regularized kernel-based methods for system identification have gained popularity in recent years. However, current formulations are not robust with respect to outliers. In this paper, we study possible solutions to robustify kernel-based methods that rely on modeling noise using the Laplacian probability density function (pdf). The contribution of this paper is two-fold. First, we introduce a new outlier robust kernel-based system identification method. It exploits the representation of Laplacian pdfs as scale mixture of Gaussians. The hyperparameters characterizing the problem are chosen using a new maximum a posteriori estimator whose solution is computed using a novel iterative scheme based on the expectation-maximization method. The second contribution of the paper is the review of two other robust kernel-based methods. The three methods are compared by means of numerical experiments, which show that all of them give substantial performance improvements compared to standard kernel-based methods for linear system identification.