Polytopal methods—numerical methods capable of handling general polygonal and polyhedral meshes—have experienced significant growth over the past decade in both the mathematics and engineering communities. Notable examples include Virtual Elements, Hybrid High-Order methods, PolyFEM, Polytopal Discontinuous Galerkin methods, and Mimetic Discretizations. These methods are particularly well-suited for addressing engineering problems in fluid and solid mechanics, owing to their remarkable flexibility in representing complex geometries, interfaces, and heterogeneous media. They also offer enhanced capabilities for mesh adaptivity due to the capability of managing efficiently hanging nodes, and elements with highly general shapes. Furthermore, polytopal methods have inspired ideas to design elements that are novel also on classical meshes, both in fluid and solid mechanics. This minisymposium aims to bring together mathematicians and engineers to discuss recent advances in polytopal methods with a particular focus on applications in mechanics.