  This paper explains the multi-orbital band structures and itinerant magnetism
of the iron-pnictide and chalcogenides. We first describe the generic band
structure of an isolated FeAs layer. Use of its Abelian glide-mirror group
allows us to reduce the primitive cell to one FeAs unit. From
density-functional theory, we generate the set of eight Fe $d$ and As $p$
localized Wannier functions for LaOFeAs and their tight-binding (TB)
Hamiltonian, $h(k)$. We discuss the topology of the bands, i.e. allowed and
avoided crossings, the origin of the d6 pseudogap, as well as the role of the
As $p$ orbitals and the elongation of the FeAs$_{4}$ tetrahedron. We then
couple the layers, mainly via interlayer hopping between As $p_{z}$ orbitals,
and give the formalism for simple and body-centered tetragonal stackings. This
allows us to explain the material-specific 3D band structures. Due to the high
symmetry, several level inversions take place as functions of $k_{z}$ or
pressure, resulting in linear band dispersions (Dirac cones). The underlying
symmetry elements are, however, easily broken, so that the Dirac points are not
protected, nor pinned to the Fermi level. From the paramagnetic TB Hamiltonian,
we form the band structures for spin spirals with wavevector $q$ by coupling
$h(k)$ and $h (k+q)$. The band structure for stripe order is studied as a
function of the exchange potential, $\Delta$, using Stoner theory. Gapping of
the Fermi surface (FS) for small $\Delta $ requires matching of FS dimensions
(nesting) and $d$-orbital characters. The origin of the propeller-shaped FS is
explained. Finally, we express the magnetic energy as the sum over
band-structure energies, which enables us to understand to what extent the
magnetic energies might be described by a Heisenberg Hamiltonian, and the
interplay between the magnetic moment and the elongation of the FeAs4
tetrahedron.
