Computing spectral properties of operators is fundamental in the sciences, with wide-ranging applications in quantum mechanics, signal processing, fluid mechanics, dynamical systems, etc. However, the infinite-dimensional problem is infamously difficult (difficulties include spectral pollution and dealing with continuous spectra). This talk aims to introduce classes of practical resolvent-based algorithms that rigorously compute a zoo of spectral properties of operators on Hilbert spaces. We also discuss how these methods form part of a broader programme on the foundations of computation (figuring out what is and what is not possible). The focus will be on two problems for general classes of discrete and differential operators: computing spectra with error control and computing spectral measures. These problems can be considered the infinite-dimensional analogues of computing eigenvalues and eigenvectors, respectively, and provide “diagonalisation” through the spectral theorem. The first problem is solved by an algorithm that approximates resolvent norms and has links with pseudospectra theory. The second is solved by computing convolutions of rational kernels with the measure via the resolvent operator (solving shifted linear systems). Numerical examples demonstrate how the new techniques can handle problems that before were out of reach. The final part of the talk provides purely data-driven algorithms that compute the spectral properties of Koopman operators (with convergence guarantees). Koopman operators “linearise” nonlinear dynamical systems, the price being a reduction to an infinite-dimensional spectral problem (c.f. “Koopmania”, describing their surge in popularity). The talk will end with applications of these new methods in several thousand state-space dimensions.