The accurate numerical approximation of partial differential equations on complex geometries remains a major challenge in scientific computing. General polygonal and polyhedral meshes, curved boundaries, heterogeneous materials, and multiphysics interfaces require discretization techniques that combine geometric flexibility, robustness, and high-order accuracy. This mini-symposium aims to highlight recent advances in high-order numerical methods, from the perspective of early-career researchers. Contributions may involve finite element, virtual element method, spectral element method, discontinuous Galerkin, hybrid-high order, isogeometric analysis, together with developments in adaptivity and refinement strategies. Particular attention will be devoted to the application of these approaches to challenging problems, including eigenvalue problems, multiscale and multiphysics models, contact and interface phenomena, and fluid–structure interaction. These applications raise fundamental questions related to accuracy, stability, and computational efficiency on complex domains. Particular emphasis will be placed on novel ideas and emerging trends, such as the interplay between standard methods and machine learning techniques, that exploit high-order discretizations to achieve accurate and efficient simulations on general computational meshes. By bringing together young scientists working across different numerical frameworks, the mini-symposium seeks to foster interactions among research communities and provide a snapshot of emerging trends and future directions on high-order computational methods for PDEs.