We present an estimator of the covariance matrix of a random d-dimensional vector from an i.i.d. finite sample. Our only assumption is that this vector satisfies a bounded L^p-L^2 marginal moment for p greater than or equal to 4, and we allow an adversary to modify an arbitrary fraction of the sample. Given this, we show that the covariance can be estimated with the same high-probability error rates that the sample covariance matrix achieves in the case of Gaussian data. This talk is based on joint work with Roberto I. Oliveira (IMPA).