  We present $\mathbf{k}\cdotp\mathbf{p}$ Hamiltonians parametrised by {\it ab
initio} density functional theory calculations to describe the dispersion of
the valence and conduction bands at their extrema (the $K$, $Q$, $\Gamma$, and
$M$ points of the hexagonal Brillouin zone) in atomic crystals of
semiconducting monolayer transition metal dichalcogenides. We review the
parametrisation of the essential parts of the $\mathbf{k}\cdotp\mathbf{p}$
Hamiltonians for MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$, including the
spin-splitting and spin-polarisation of the bands, and we discuss the
vibrational properties of these materials. We then use
$\mathbf{k}\cdotp\mathbf{p}$ theory to analyse optical transitions in
two-dimensional transition metal dichalcogenides over a broad spectral range
that covers the Van Hove singularities in the band structure (the $M$ points).
We also discuss the visualisation of scanning tunnelling microscopy maps.
