Adaptive methods are essential for approximating the solutions of partial differential equations in several engineering applications. These methods rely on a posteriori error estimators, which identify regions where the error is most concentrated, guiding targeted refinements. In this context, Virtual Element Methods (VEMs) offer a compelling approach, as they admit the presence of hanging nodes, allowing localized refinement. However, standard VEM formulations incorporate arbitrary bilinear forms to stabilize the scheme. These stabilization terms also appear on the right-hand side of classical a posteriori error estimates, posing many challenges. To address this issue, we propose two strategies. First, building on the approach proposed in [1], we derive a stabilization-free a posteriori error estimator for arbitrary orders in the case of triangular [2] and tetrahedral shape meshes [3], allowing hanging nodes. This leads to prove a contraction result, and then to define an adaptive algorithm for some classes of meshes. The second strategy focuses on the a posteriori error analysis for the Stabilization Free Virtual Element Method (SFVEM). By introducing new polynomial projections, we establish the equivalence between a suitably defined error measure and classical residual error estimators [4]. REFERENCES [1] L. Beirao da Veiga, C. Canuto, R. H. Nochetto, G. Vacca, G. and M. Verani, Adaptive VEM: Stabilization-Free A Posteriori Error Analysis and Contraction Property. SIAM J. Numer. Anal., Vol. 61 (2), pp. 457–494, 2023. [2] C. Canuto and D. Fassino, Higher-order adaptive virtual element methods with contraction properties. Math. Eng., Vol. 5 (6), pp. 1–33, 2023. [3] S. Berrone, D. Fassino, F. Vicini, 3D Adaptive VEM with Stabilization-Free a Posteriori Error Bounds. Journal of Scientific Computing, 2025, 103, 35. [4] S. Berrone, A. Borio, D. Fassino and F. Marcon, A residual a posteriori error estimate for the Stabilization-free Virtual Element Method. Submitted for publication, 2025.