[en] We present $\mathbf{k}\cdot <!--\; ???\; -->\mathbf{p}$ Hamiltonians parametrized by ab initio density functional theory calculations to describe the dispersion of the valence and conduction bands at their extrema (the K, Q, Γ, and M points of the hexagonal Brillouin zone) in atomic crystals of semiconducting monolayer transition metal dichalcogenides (TMDCs). We discuss the parametrization of the essential parts of the $\mathbf{k}\cdot <!--\; ???\; -->\mathbf{p}$ Hamiltonians for MoS₂, MoSe₂, MoTe₂, WS₂, WSe₂, and WTe₂, including the spin-splitting and spin-polarization of the bands, and we briefly review the vibrational properties of these materials. We then use $\mathbf{k}\cdot <!--\; ???\; -->\mathbf{p}$ theory to analyse optical transitions in two-dimensional TMDCs over a broad spectral range that covers the Van Hove singularities in the band structure (the M points). We also discuss the visualization of scanning tunnelling microscopy maps. (topical review)