Chapter 1 shows why spectral computation is inherently infinite-dimensional. It shows how standard discretisations (finite sections, domain truncation) can fail even for normal operators, through spectral pollution (spurious limit points), spectral invisibility (missing spectrum), essential/continuous spectrum, and ill-conditioning. Concrete examples are given from magnetic Schrödinger and quasicrystal models, convection–diffusion, and a Laurent ‘whale’ operator. The chapter formulates the classical computational spectral problem for operators given by matrix entries, and introduces Hausdorff and Attouch–Wets convergence as the right notion of 'no pollution/no invisibility'. It presents a three-limit convergent algorithm based on injection moduli (smallest singular values) and their link to the resolvent. Finally, it motivates the Solvability Complexity Index (SCI) hierarchy and the resolvent as algorithmic tools, with warm-up classifications (diagonal and Jacobi operators) and an impossibility result for verified error control of general self-adjoint compact operators.