In this talk, we present recent results on the stability of three volume-preserving geometric flows in the N-dimensional flat torus: the volume-preserving Mean Curvature Flow, the Surface Diffusion Flow, and the Mullins-Sekerka Flow. Specifically, we show that, for initial data sufficiently close to a strictly stable critical set of the perimeter (in an appropriate sense), these flows exist for all times and converge exponentially fast to a translate of the stable set as the time tends to infinity. This work has been done in collaboration with Daniele De Gennaro, Antonia Diana and Andrea Kubin.