In the last few years, the support vector machine (SVM) method has motivated new interest in kernel regression techniques. Although the SVM has been shown to exhibit excellent generalization properties in many experiments, it suffers from several drawbacks, both of a theoretical and a technical nature: the absence of probabilistic outputs, the restriction to Mercer kernels, and the steep growth of the number of support vectors with increasing size of the training set. In this paper, we present a different class of kernel regressors that effectively overcome the above problems. We call this approach generalized LASSO regression. It has a clear probabilistic interpretation, can handle learning sets that are corrupted by outliers, produces extremely sparse solutions, and is capable of dealing with large-scale problems. For regression functionals which can be modeled as iteratively reweighted least-squares (IRLS) problems, we present a highly efficient algorithm with guaranteed global convergence. This defies a unique framework for sparse regression models in the very rich class of IRLS models, including various types of robust regression models and logistic regression. Performance studies for many standard benchmark datasets effectively demonstrate the advantages of this model over related approaches.