Mesh generation in highly complex domains remains one of the most challenging open problems in numerical analysis. In this talk, we address a posteriori error estimates, adaptive mesh refinement, and mesh generation within the framework of polytopal methods [1,4]. 2D and 3D polytopal methods often require stabilization terms, which can pose significant challenges for a posteriori error estimates. However, recent developments in Virtual Element Methods (VEM) and residual-based a posteriori error estimates have shown that stabilization terms can be bounded by other residual terms, provided that triangular and tetrahedral meshes with a bounded number of hanging nodes are used [5]. The recently introduced stabilization-free VEM offers a promising strategy to overcome this issue [2,3]. This talk will also present a novel approach for refining convex polygonal meshes designed to preserve or enhance mesh quality during refinement [1,4]. After several iterations, this process converges to high-quality triangular or quadrilateral meshes. Starting from a very coarse mesh generated solely from the domain description, the proposed method can be interpreted as an efficient tool for generating high-quality polygonal meshes. References [1] Berrone S., D'Auria A., (2022), A new quality preserving polygonal mesh refinement algorithm for Polygonal Element Methods, Finite Elements in Analysis and Design, vol. 207. [2] Berrone S.; Borio A.; Marcon F., (2024), A stabilization-free Virtual Element Method based on divergence-free projections, Computer Methods in Applied Mechanics and Engineering, vol. 424, pp. 1-19. [3] Berrone S.; Borio A.; Marcon F., (2025), Lowest order stabilization free virtual element method for the 2D Poisson equation, Computers & Mathematics with Applications. [4] Berrone S., Vicini F., (2025), Effective polygonal mesh generation and refinement for VEM, Mathematics and Computers in Simulation. [5] Berrone S., Fassino D., Vicini F., (2025), 3D Adaptive VEM with stabilization-free a posteriori error bounds, arXiv:2407.17858.