The goal of this text is to review recent advances and to present new results in the numerical analysis of the finite sections method for general band and band-dominated operators. The main topics are the stability of the finite sections method and the asymptotic behavior of singular values. The latter topic is closely related with compactness and Fredholm properties of approximation sequences, and the paper can also serve as an introduction into this remarkable field of numerical analysis. Further the author discusses the behavior of approximation numbers, determinants, essential spectra and essential pseudospectra as well as the localization of pseudomodes of finite sections of band-dominated operators.