The observation at the LHC of a scalar particle compatible with the Standard Model (SM) Higgs boson could represent the ultimate success of the SM description of particle interactions. At the same time, the lack of significant deviations from the SM picture, both in direct searches at high energy colliders and in electroweak and flavor precision observables, seems to exclude the presence of new physics at scales lower than a few TeV. Actually, a 125 GeV SM Higgs boson allows the SM to be consistently extrapolated, together with all its known conceptual issues, up to the Planck scale: even if absolute stability of the radiatively corrected potential is disfavored, we are likely to reside in a metastable minimum, whose lifetime turns out to be much longer than the age of the universe. In other words, given the current knowledge of the SM parameters, the need for new physics below the Planck scale (where gravity is expected to become relevant) cannot be justified on the basis of sole vacuum stability considerations. In this talk I will first review the concepts that are relevant for the study of the SM vacuum stability. Then I will discuss the state-of-the-art analysis and some of its implications.