In this talk, we present an adaptive virtual element method on general polytopal meshes for the numerical approximation of fourth-order partial differential equations. The approach relies on a conforming $C^1$ virtual element discretisation, which naturally handles arbitrary polygonal and polyhedral meshes and is well suited for complex geometries. A central component of the method is a residual-based a posteriori error estimator that drives an adaptive mesh refinement strategy. We prove reliability and efficiency of the estimator within a dimension independent framework, ensuring robustness in both two and three dimensions. Numerical experiments in two and three dimensions, including benchmark eigenvalue problems and applications to wind-driven ocean circulation models, demonstrate optimal convergence rates and effective error control.