  We provide an estimator of the covariance matrix that achieves the optimal
rate of convergence (up to constant factors) in the operator norm under two
standard notions of data contamination: We allow the adversary to corrupt an
$\eta$-fraction of the sample arbitrarily, while the distribution of the
remaining data points only satisfies that the $L_{p}$-marginal moment with some
$p \ge 4$ is equivalent to the corresponding $L_2$-marginal moment. Despite
requiring the existence of only a few moments, our estimator achieves the same
tail estimates as if the underlying distribution were Gaussian. As a part of
our analysis, we prove a dimension-free Bai-Yin type theorem in the regime $p >
4$.
