Spectral computation in infinite dimensions is fundamentally different from its finite-dimensional counterpart. Standard discretisation strategies, including finite sections, domain truncation, and eigenvalue solvers, can fail spectacularly, producing spectral pollution, spectral invisibility, and ill-conditioned limits even for normal or self-adjoint operators. This talk presents a unified framework that brings together foundational results and practical algorithms for understanding and overcoming these failures. The central theme is a resolvent-based, infinite-dimensional perspective on spectral computation, coupled with the Solvability Complexity Index (SCI) hierarchy, which precisely characterises what spectral information can be computed, how many limiting processes are required, and when certified error control is impossible. I will explain why multi-limit algorithms are unavoidable, how injection moduli and pseudospectra lead to pollution-free and invisible-free methods, and how these ideas extend beyond spectra to spectral measures, spectral types, essential spectra, and nonlinear operator families. The programme is illustrated, where time permits, with examples ranging from Schrödinger and Dirac operators to quasicrystals, non-normal stability problems, and data-driven Koopman operators, highlighting both sharp impossibility results and practical, provably convergent algorithms.
