The rigidity of the Riemannian positive mass theorem for asymptotically flat or hyperbolic manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the Euclidean space, or hyperbolic space, respectively. This leads us to ask us wether a stability result holds. We will provide a positive answer obtained by Huang-Lee-Sormani, Allen-Perales and Huang-Lee-Perales for asymptotically flat graphical manifolds by using the intrinsic flat distance. We will also discuss an analogous result for asymptotically hyperbolic graphical manifolds, which is part of an ongoing project with A. Cabrera Pacheco and Melanie Graf.