This minisymposium focuses on state-of-the-art numerical methods for PDE eigenvalue problems, central in scientific computing due to their critical importance in stability analysis, vibration modeling, quantum physics, etc. Presentations will highlight novel discretization and approximation techniques for standard and nonstandard PDE eigensolvers, as well as approaches for nonlinear eigenvalue problems. Some time will also be dedicated to talks on outer-loop methodologies designed to handle parametric and stochastic eigenvalue frameworks such as those arising in inverse problems, optimal design, and uncertainty quantification.