For more than a decade, Virtual Element Methods (VEMs) have provided an efficient framework for solving partial differential equations on arbitrary polyhedral meshes. Standard VEM formulations rely on bilinear stabilization terms; however, the arbitrary and non-polynomial character of these terms may pose challenges in the a posteriori error analysis. In particular, stabilization terms also appear on the right-hand side of classical a posteriori error estimates. In this talk, we extend the analysis presented in [1], which establishes stabilization-free a posteriori error bounds for lowest-order VEM discretizations on triangulations with hanging nodes, to more general settings [5, 4]. Furthermore, with the aim of deriving stabilization-free a posteriori error bounds for general polygonal meshes, we present the Stabilization-free Virtual Element Method [2]. The introduction of novel polynomial projection operators enables the construction of structure-preserving schemes and allows us to prove the equivalence between a suitably defined error measure and standard residual-based error estimators [3]. REFERENCES [1] Beirão da Veiga, L., Canuto, C., Nochetto, R. H., Vacca, G., Verani, M.,Adaptive VEM: Stabilization-Free A Posteriori Error Analysis and Contraction Property, SIAM Journal on Numerical Analysis, 61(2), 457–494, 2023 [2] Berrone S., Borio A., Fassino D., Marcon F., Stabilization-free Virtual Element Method for 2D second order elliptic equations, Computer Methods in Applied Mechanics and Engineering, 438, 117839, 2025. [3] Berrone S., Borio A., Fassino D., Marcon F., A residual a posteriori error estimate for the Stabilization-free Virtual Element Method, Submitted for publication. [4] Berrone S., Fassino D., Vicini F., 3D Adaptive VEM with Stabilization-Free a Posteriori Error Bounds, Journal of Scientific Computing, 103, 35, 2025. [5] Canuto, C., Fassino D., Higher-order adaptive virtual element methods with contraction properties, Mathematics in Engineering, 5(6), 1-33, 2023.