  Twisted bilayers of two-dimensional materials, such as twisted bilayer
graphene, often feature flat electronic bands that enable the observation of
electron correlation effects. In this work, we study the electronic structure
of twisted transition metal dichalcogenide (TMD) homo- and heterobilayers that
are obtained by combining MoS$_2$, WS$_2$, MoSe$_2$ and WSe$_2$ monolayers, and
show how flat band properties depend on the chemical composition of the bilayer
as well as its twist angle. We determine the relaxed atomic structure of the
twisted bilayers using classical force fields and calculate the electronic band
structure using a tight-binding model parametrized from first-principles
density-functional theory. For homobilayers, we find that the two highest
valence bands exhibit a graphene-like dispersion and become flat as the twist
angle is reduced. In contrast, not all heterobilayers have flat valence bands.
Specifically, we find that those systems in which the highest valence band
derives from K or K' states of the constituent monolayers do not exhibit flat
bands, even at small twist angles. In all systems, qualitatively different band
structures are obtained when atomic relaxations are neglected.
