Numerical methods for Partial Differential Equations based on general polygonal and polyhedral meshes represent a highly active and evolving research field. Their success is primarily driven by their inherent flexibility in handling complex geometries, which supports seamless adaptive mesh refinement and enables the construction of discrete spaces with tailored physical or mathematical properties. This minisymposium aims to mix rigorous mathematical foundations and high-per\-for\-man\-ce computational applications. We welcome contributions on a wide range of polytopal frameworks, such as the Virtual Element Method, Hybrid High-Order methods, and Discontinuous Galerkin schemes, focusing on both their theoretical properties and their impact on complex simulations. Key topics include advanced error analysis (a priori and a posteriori), stability, and the preservation of physical structures (e.g., divergence-free constraints or tensor symmetry) at the discrete level. Simultaneously, the session will explore the practical challenges arising in computational mechanics, fluid dynamics, and multiphysics. Particular interest is also devoted to the design of efficient preconditioners, scalable solvers, and parallel implementations, which are essential for transitioning these modern discretizations from theoretical prototypes to robust tools for large-scale scientific computing.