I review some applications of random matrix theory to multivariate statistics. Suppose one is interested in the covariance matrix of a random vector whose distribution is unknown. In order to determine the covariances from empirical observations, one approximates them using empirical averages obtained from a series of measurements. The resulting sample covariance matrix is random, and its relationship with the true covariance matrix rather intricate. I outline some recent progress in understanding the behaviour of sample covariance matrices. The cornerstone of the proofs is an anisotropic local law for the resolvent. Applications include the Tracy-Widom distribution of eigenvalues near the spectral edges.