Experiments indicate that the linear, low bias, conductance of devices made by long insulating molecular chains connected to metallic electrodes decays exponentially with the number n of monomers in the molecular chain, according to a simple tunneling law which depends on two parameters, the decay constant and the pre-exponential factor . Understanding how , the contact conductance, and depend on the properties of the molecule, of the electrodes, and of the contacts is an issue of relevant current interest.