A key issue in the context of (compact) extra dimensions is the one of their stability. Any stabilization mechanism is effective only up to some given energy scale; if they can approach this energy, 4$d observers can excite the fluctuations of the internal space, and probe its existence. Stabilization mechanisms introduce fields in the internal space; perturbations of these fields are mixed with perturbations of the metric, so that their study requires a complete GR treatment. After presenting the general framework, I will then discuss some relevant applications. I will present the exact coupling of the radion to codimension one branes, extending the regime of validity of the results in the literature. I will then focus on de Sitter compactifications, showing that the cosmological expansion has typically the effect of destabilizing the internal space. The final part of the talk will be devoted to related work in progress in less conventional areas of brane models: the localization of gravity towards the IR brane (corresponding to a dual description of emerging gravity from the CFT), and the inclusion of ghost fields in the bulk.