I describe recent results on spiked covariance matrices, which model multivariate data containing nontrivial correlations. In principal component analysis, one extracts the leading contribution to the covariance by analysing the top eigenvalues and associated eigenvectors of the covariance matrix. I give non-asymptotic, sharp, high-probability estimates relating the principal components of the true covariance matrix to those of the sample covariance matrix. In particular, I discuss the behaviour of eigenvalues and eigenvectors at and near the so-called BBP phase transition, at which outlier eigenvalues are created or annihilated. Joint work with Alex Bloemendal, Horng-Tzer Yau, and Jun Yin.