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Mathematicians, physicists, engineers, and data scientists will welcome this comprehensive, rigorous, and practical guide to computing spectral properties of operators in infinite-dimensional settings. It explains why standard discretisation can fail and shows how to overcome these pitfalls. It develops resolvent-based algorithms with provable convergence and certified error bounds, organised by a precise computability classification that clarifies what is achievable, what is impossible, and what extra information makes problems tractable. Topics include spectra and pseudospectra, spectral measures and functional calculus, spectral types, fractal and Cantor-type spectra, essential versus discrete spectra and multiplicities, spectral radii, abscissas and gaps, nonlinear operator pencils, and verified computation. A distinctive feature is the integration of modern applications, including a fully rigorous treatment of data-driven Koopman spectral analysis. Hundreds of worked examples, exercises with solutions, notes, and usable code make the book both a reference and a powerful toolkit for researchers and students.