Latent variable models lay the statistical foundation for data science problems with unstructured, incomplete and heterogeneous information. Spectral methods extract low-dimensional geometric structures for downstream tasks in a computationally efficient way. Despite their conceptual simplicity and wide applicability, theoretical understanding is lagging far behind and that hinders development of principled approaches. In this talk, I will first talk about the bias and variance of PCA, and apply the results to distributed estimation of principal eigenspaces. Then I will present an $ell_p$ theory of eigenvector analysis that yields optimal recovery guarantees for spectral methods in many challenging problems. The results find applications in dimensionality reduction, mixture models, network analysis, recommendation systems, ranking and beyond.