Adaptive high-order methods provide a powerful framework for the accurate and efficient numerical approximation of partial differential equations and associated eigenvalue problems, particularly in the presence of singularities, localized features, complex geometries, or multiple spatial scales. This minisymposium aims to bring together young researchers working on theoretical, computational, and practical aspects of adaptive numerical methods across a broad range of approximation frameworks, including finite element methods, space–time approaches, and other high-order techniques. Particular emphasis will be placed on a posteriori error estimation, reliability and efficiency analysis, convergence and optimality of adaptive algorithms, and mesh and polynomial degree refinement strategies. By presenting complementary approaches and recent developments, the minisymposium seeks to promote scientific exchange among early-career researchers, strengthen existing scientific connections, and foster new collaborations.