This work aims to study the stability of minimally immersed hypersurfaces in R n more 1. We present some characterizations of minimal hypersurfaces deducting the formulas of the first and second variation of area. Afterwards, from the variational calculus, we establish the relationship between spectral theory and stability. Particulary, we study a variational characterization of the first eigenvalue associated to the stability operator. Based in this relationship we show some stability criteria for minimally immersed hypersurfaces in R n more 1. In particular, we exhibit in details the Barbosa-Do Carmo criterion for the stability of minimal surfaces in R3. We also establish the Fischer- Colbrie-Shoen criterion for complete, non compact, minimal surfaces using the elliptic theory. We conclude with the analysis of the stability of the catenoid in R3 and in Rn more 1. This is done by studying the stability domains of the catenoid in R3 using the Sturm-Liouville theory. We explain the Lindelof stability theorem in R3 and in R n more 1 and the property of the catenoids have index 1.