  In gauge theories, observable quantities have to be gauge-invariant. In
general, this requires composite operators, which usually have substantially
different properties, e.g. masses, than the elementary particles. Theories with
a Higgs field, in which the Brout-Englert-Higgs effect is active, provide an
interesting exception to this rule. Due to an intricate mechanism, the
Fr\"ohlich-Morchio-Strocchi mechanism, the masses of the composite operators
with the same $J^P$ quantum numbers, but modified internal quantum numbers,
have the same masses. This mechanism is supported using lattice gauge theory
for the standard-model Higgs sector, i.e. Yang-Mills-Higgs theory with gauge
group SU(2) and custodial symmetry group SU(2). Furthermore, the extension to
the 2-Higgs-doublet-model is briefly discussed, and some preliminary results
are presented.
