I will present a general method (pioneered by Ros and Savo) to obtain universal and effective index estimates for minimal hypersurfaces inside a Riemannian manifold, given an isometric embedding of the latter in some (possibly high-dimensional) Euclidean space. This approach can be applied, on the one hand, to tackle a conjecture by Schoen and Marques-Neves asserting that the Morse index of a closed minimal hypersurface in a manifold of positive Ricci curvature is bounded from below by a linear function of its first Betti number, which we settle for a large class of ambient spaces. On the other hand, these methods turn out to be very powerful in studying free-boundary minimal hypersurfaces in Euclidean domains: among other things, we prove a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Applications to compactness theorems, to the explicit analysis of known examples (due to Fraser-Schoen and to Folha-Pecard-Zolotareva) and to novel classification theorems will also be mentioned. This is joint work with Lucas Ambrozio and Benjamin Sharp.