  The current searches for a heavy Higgs boson assume on-shell (stable)
Higgs-boson production. The Higgs-boson production cross section is then
sampled with a Breit-Wigner distribution (with fixed-width or running-width)
and implemented in MonteCarlo simulations. Therefore the question remains of
what is the limitation of the narrow Higgs-width approximation. The main focus
of this work is on the description of the Standard Model Higgs--boson-lineshape
in the heavy Higgs region, typically $\MH$ above $600\UGeV$. The framework
discussed in this paper is general enough and can be used for all processes and
for all kinematical regions. Numerical results are shown for the gluon-fusion
process. Issues of gauge invariance and residual theoretical uncertainties are
also discussed. Limitations due to a breakdown of the perturbative expansion
are comprehensively discussed, including a discussion of th equivalence theorem
for (off-shell) virtual vector-bosons. Analytic continuation in a theory with
unstable particles is thoroughly discussed.
