Extreme value theory for univariate and low-dimensional observations has been explored in considerable detail, but the field is still in an early stage regarding high-dimensional settings. This paper focuses on Hüsler-Reiss models, a popular class of models for multivariate extremes similar to multivariate Gaussian distributions, and their domain of attraction. We develop estimators for the model parameters based on score matching, and we equip these estimators with theories and exceptionally scalable algorithms. Simulations and applications to weather extremes demonstrate the fact that the estimators can estimate a large number of parameters reliably and fast; for example, we show that Hüsler-Reiss models with thousands of parameters can be fitted within a couple of minutes on a standard laptop. More generally speaking, our work relates extreme value theory to modern concepts of high-dimensional statistics and convex optimization.