In this paper, we propose a novel method for Support Vector Regression (SVR) based on second-order cones. The proposed approach defines a robust worst-case framework for the conditional densities of the input data. Linear and kernel-based second-order cone programming formulations for SVR are proposed, while the duality theory allows us to derive interesting geometrical properties for this strategy: the method maximizes the margin between two ellipsoids obtained by shifting the response variable up and down by a fixed parameter. Experiments for regression on twelve well-known datasets confirm the superior performance of our proposal compared to alternative methods such as standard SVR and linear regression. (C) 2018 Elsevier B.V. All rights reserved.