Spectral geometry concerns the interplay between the shape of a domain or manifold and the spectrum of an associated operator, such as the Laplace–Beltrami operator or the Dirichlet-to-Neumann (Steklov) map. In case of Extremal questions — which shapes optimize a given eigenvalue — connect analysis, geometry, and computation, with recent applications to free boundary minimal surfaces, sharp spectral inequalities, and computer-assisted proofs. Progress on these problems is often challenged by discretization accuracy: eigenvalue optimization demands many eigenpairs computed to high precision on evolving, non-smooth, or high-genus geometries. If the operators are vectorial or if the domain is unbounded, these questions present additional challenges. High-order methods — boundary integral and layer-potential formulations, the method of particular solutions and related fundamental solution approaches, spectral schemes, hp- and isogeometric finite elements — are uniquely suited to deliver, and increasingly certify, this accuracy. This minisymposium brings together researchers developing high-order discretizations for spectral problems in challenging settings with those applying them to optimization and inverse problems in spectral geometry.