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Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues (resp. singular values) and eigenvectors (resp. singular vectors) of some properly designed matrices constructed from data. A diverse array of applications have been found in machine learning, imaging science, financial and econometric modeling, and signal processing, including recommendation systems, community detection, ranking, structured matrix recovery, tensor data estimation, joint shape matching, blind deconvolution, financial investments, risk managements, treatment evaluations, causal inference, amongst others. Due to their simplicity and effectiveness, spectral methods are not only used as a stand-alone estimator, but also frequently employed to facilitate other more sophisticated algorithms to enhance performance.