High-order polytopal methods have emerged as a powerful framework for the numerical approximation of partial differential equations (PDEs), combining high-order accuracy with the flexibility of general polygonal and polyhedral meshes. This mini-symposium, organized in the framework of the ERC SyG NEMESIS project, brings together researchers working on the analysis, development, and application of advanced, high-order, polytopal discretization techniques, including Virtual Elements, Hybrid High-Order, (hybridized) polytopal discontinuous Galerkin, Discrete De Rham methods, and related approaches. Contributions will address theoretical advances, efficient algorithms, adaptivity, machine-learning enhancement, and applications to challenging problems in science and engineering. The session aims to foster exchanges between experts in numerical analysis, scientific computing, and applied mathematics, highlighting recent developments and future directions in high-order polytopal methods.