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Suppose we are observing a sample of independent random vectors, knowing that the original distribution was contaminated, so that a fraction of observations came from a different distribution. How to estimate the covariance matrix of the original distribution in this case? In this talk, we discuss an estimator of the covariance matrix that achieves the optimal dimension-free rate of convergence under two standard notions of data contamination: We allow the adversary to corrupt a fraction of the sample arbitrarily, while the distribution of the remaining data points only satisfies a certain (rather weak) moment equivalence assumption. Despite requiring the existence of only a few moments, our estimator achieves the same tail estimates as if the underlying distribution were Gaussian. Based on a joint work with Pedro Abdalla.