Conduction properties of nanoscale contacts can be studied using first-principles simulations. Such calculations give insight into details behind the conductance that is not readily available in experiments. For example, we may learn how the bonding conditions of a molecule to the electrodes affect the electronic transport. Here we describe key computational ingredients and discuss these in relation to simulations for scanning tunneling microscopy (STM) experiments with C60 molecules where the experimental geometry is well characterized. We then show how molecular dynamics simulations may be combined with transport calculations to study more irregular situations, such as the evolution of a nanoscale contact with the mechanically controllable break-junction technique. Finally we discuss calculations of inelastic electron tunnelling spectroscopy as a characterization technique that reveals information about the atomic arrangement and transport channels.