We present a numerical study of the electronic structure of simple models of liquid and amorphous selenium. The geometry is based upon random walks; the electronic structure is described by a tight-binding Hamiltonian. We investigate the influence of the bond- and dihedral-angle distribution, the range and nature of the tight-binding hopping matrix element and the sign and magnitude of the electron - electron interaction upon the density of states. Localization properties and charge profiles have been computed. To reproduce the experimentally observed optical gap, it is essential to fix the dihedral angles and to introduce an intraorbital electron - electron attraction of modest strength . In contrast to local descriptions of bond breaking in disordered Se, we observe the formation of two negatively charged chain ends, compensated by the creation of a bipolaron within the chain. Only bipolaron states are able to occupy an impurity band. A simple mechanism for Fermi level pinning is discussed. We give an outlook on the electronic properties of related three-dimensional models.