This mini-symposium focuses on recent advances in the fundamentals of spectral methods, including new transforms for univariate and multivariate orthogonal polynomials, fast matrix decompositions and structured algorithms for systems arising from spectral discretisation, methods for fractional-order equations, approximation theory for spectral expansions, and novel spectral methods on non-standard and unbounded domains. Further topics include spectral approximation of singular functions, scaling strategies for Hermite and Laguerre methods, sparse spectral methods on geometries described by algebraic curves and surfaces, and stable high-order time-stepping schemes. The goal is to provide a platform for exchanging ideas, identifying common analytical and computational challenges, fostering collaboration, and advancing the state of the art in spectral methods.