  The discovery of a Higgs particle has triggered numerous theoretical and
experimental investigations concerning its production and decay rates and has
led to interesting results concerning its interaction with fermions and gauge
bosons. The self-interaction $\lambda$ of the Standard Model Higgs boson is
particularly important due to its close connection with the stability of the SM
vacuum. In this talk precision calculations for the evolution of this crucial
coupling are presented and their impact on the question of vacuum stability is
analysed. We also compare the theoretical precision resulting from the
calculation of three-loop $\beta$-functions to the experimental uncertainties
stemming from key parameters, such as the top mass, the Higgs mass and the
strong coupling, and to the theoretical uncertainties introduced by the
matching of experimental data to parameters in the theoretically favoured
$\overline{\text{MS}}$ renormalization scheme.
