Abstract Multivariate multiple linear regression is a widely used statistical technique for modelling relationships between some response variables and several predictor variables. Traditional likelihood-based methods can produce very misleading results in the presence of outliers. In this work, we propose two robust multivariate regression methods designed to handle high-dimensional data: one based on the minimum regularized covariance determinant estimator, a robust estimator of location and scatter for high-dimensional data; and another based on dimensionality reduction using robust sparse principal component analysis. Through a study simulation, we evaluate the robustness and efficiency of the estimators obtained, the ability of the methodologies to correctly classify observations in contaminated datasets, and the computational cost. A real data application illustrates the use of the proposed methodologies.