  The semiconducting two-dimensional transition metal dichalcogenides MX$_{2}$
show an abundance of one-dimensional metallic edges and grain boundaries.
Standard techniques for calculating edge states typically model nanoribbons,
and require the use of supercells. In this paper we formulate a Green$'$s
function technique for calculating edge states of (semi-)infinite
two-dimensional systems with a single well-defined edge or grain boundary. We
express Green$'$s functions in terms of Bloch matrices, constructed from the
solutions of a quadratic eigenvalue equation. The technique can be applied to
any localized basis representation of the Hamiltonian. Here we use it to
calculate edge states of MX$_{2}$ monolayers by means of tight-binding models.
Besides the basic zigzag and armchair edges, we study edges with a more general
orientation, structurally modifed edges, and grain boundaries. A simple
three-band model captures an important part of the edge electronic structures.
An eleven-band model comprising all valence orbitals of the M and X atoms, is
required to obtain all edge states with energies in the MX$_{2}$ band gap. Here
states of odd symmetry with respect to a mirror plane through the layer of M
atoms have a dangling-bond character, and tend to pin the Fermi level.
