The structure and dynamics of the fast-ionic conductor Na+-beta -alumina are studied by classical molecular dynamics simulation. Four differentcompounds are simulated: stoichiometric-, non-stoichiometric- and two Mg-stabilised beta -aluminas. The Beevers-Ross (BR)-site is the most stable position for stoichiometric Na+ ions in the conduction plane. Excess Na+ ions are found to occupy positions that are slightly off-center from the aBR lattice site. This shifted aBR, or A-site, occupation is associated with a reconstruction of three ions and a neighbouring BR vacancy. Interstitial oxygens in the conduction plane stabilise the A-site by creating permanent BR vacancies. The diffusion coefficients and conductivities as a function of temperature display close to Arrhenius behaviour. Results are in agreement with experiment for the 300-900 K regime. We find a tendency for an increasing apparent activation energy at higher temperatures.