The idea behing the quantum network (QN) model is simple enough. One joins each atom to its neighbours, and then treat electron (though quantum mechanically of course) as though they flowed through one-dimensional wires as in an electrical circuit obeying Kirfhhoff's Laws at every node. Here we will begin with two periodic systems: namely an infinite graphene layer and a two-dimensional sheet of boson atoms. This will be followed by a discussion of C and especially B nanotubes. Finally a brief summary will be given on the nature of the electronic states in a disorderd network, via an approximate treatment related to Boltzmann's equation.