  The aim of this review is to describe the field-theoretical foundations of
Brout-Englert-Higgs (BEH) physics, and to show how the usual phenomenology
arises from it. This requires to give a precise and gauge-invariant meaning to
the underlying physics. This is complicated by the fact that concepts like the
Higgs vacuum expectation value or the separation between confinement and the
BEH effect loose their meaning beyond perturbation theory. This is addressed by
carefully constructing the corresponding theory space and the quantum phase
diagram. The physical spectrum needs then to be also given in terms of
gauge-invariant, i. e. composite, states. Using gauge-invariant perturbation
theory, as developed by Froehlich, Morchio, and Strocchi, it is possible to
rederive conventional perturbation theory. This derivation explicitly shows why
the description of the standard model in terms of the unphysical,
gauge-dependent, elementary states of the Higgs and W-bosons and Z-boson, but
also of the elementary fermions, is adequate and successful. These are
unavoidable consequences of the field theory underlying the standard model,
from which the usual picture emerges. The validity of this emergence can only
be tested non-perturbatively. Such tests, in particular using lattice gauge
theory, will be reviewed as well. They fully confirm the underlying mechanisms.
It will be seen that the structure of the standard model is very special, and
qualitative changes occur beyond it. The extension beyond the standard model
will therefore also be reviewed. Particular attention will be given to
structural differences arising for phenomenology. Again, non-perturbative tests
of these results will be reviewed. Finally, to make this review self-contained
a brief discussion of issues like the triviality and hierarchy problem, and how
they fit into a fundamental field-theoretical formulation, is included.
