  Since the discovery of a Higgs particle the effective Higgs potential of the
Standard Model or extensions and the stability of the ground state
corresponding to its minimum at the electroweak scale have been subject to a
lot of investigation. The vacuum expectation value of the scalar SU(2) doublet
field in the Standard Model, which is responsible for the masses of elementary
particles, may in fact not be at the global minimum of the effective Higgs
potential. The question whether there is a deeper minimum at some large scale
is closely linked to the behaviour of the running quartic Higgs
self-interaction $\lambda(\mu)$. In this talk an update on the analysis of the
evolution of this coupling is given. We use three-loop beta-functions for the
Standard Model couplings, two-loop matching between on-shell and MSbar
quantities and compare the theoretical precision achieved in this way to the
precision in the latest experimental values for the key parameters.
