  Semimetals, in which conduction and valence bands touch but do not form Fermi
surfaces, have attracted considerable interest for their anomalous properties
starting with the discovery of Dirac matter in graphene and other
two-dimensional honeycomb materials. Here we introduce a family of
three-dimensional honeycomb systems whose electronic band structures exhibit a
variety of topological semimetals with Dirac nodal lines. We show that these
nodal lines appear in varying numbers and mutual geometries, depending on the
underlying lattice structure. They are stabilized, in most cases, by a
combination of time-reversal and inversion symmetries and are accompanied by
topologically protected "drumhead" surface states. In the bulk, these nodal
line systems exhibit Landau level quantization and flat bands upon applying a
magnetic field. In the presence of spin-orbit coupling, these topological
semimetals are found to generically form (strong) topological insulators. This
comprehensive classification of the electronic band structures of
three-dimensional honeycomb systems might serve as guidance for future material
synthesis.
