  We consider the problem of a proper definition of the mass of a heavy
unstable boson. It is shown how various definitions are related by the
renormalization-scheme independence of the full resummed amplitude. This is
made explicit for the $W$ boson in the large $N_f$ limit, where we find an
upper bound on the peak position of the lineshape and on the mass, provided a
sensible definition of the mass is used. In more realistic models, we
conjecture that under certain conditions the mass is similarly bounded. Using
this approach an upper limit of about 1 TeV on the mass of the
Minimal-Standard-Model Higgs boson is found, independent of any breakdown of
perturbation theory.
