We aim to introduce and discuss some concepts and results on the stability of minimal submanifolds, mainly when the ambient manifold is conformal either to a round sphere or to a convex domain of the Euclidean space. In particular, we will talk about a joint work with Alcides de Carvalho and Federico Trinca, in which we showed that a conformally Euclidean manifold with convex boundary does not admit volume-minimizing free boundary minimal submanifolds.