  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. We find that the highest valence bands in these
systems can derive either from $\Gamma$-point or $K$/$K'$-point states of the
constituent monolayers. For homobilayers, the two highest valence bands are
composed of monolayer $\Gamma$-point states, exhibit a graphene-like dispersion
and become flat as the twist angle is reduced. The situation is more
complicated for heterobilayers where the ordering of $\Gamma$-derived and
$K$/$K'$-derived states depends both on the material composition and also the
twist angle. In all systems, qualitatively different band structures are
obtained when atomic relaxations are neglected.
