We have reviewed the work on molecular dynamics simulation of transport properties of liquid crystals during the last 25 years. To begin with, we present the model systems that have been used in molecular dynamics simulation of liquid crystals, such as the hard sphere fluid, the Gay-Berne fluid and atomistic models. Then we explain the theory necessary for obtaining numerical estimates of the viscosities. They fall into three classes: evaluation of equilibrium fluctuation relations or Green-Kubo relations, nonequilibrium methods such as shear flow simulations and approximate methods based on the rotational diffusion coefficient. Then we review various simulation results for the Miesowicz viscosities which are of interest for lubrication and the twist viscosity, which is the technologically most important viscosity because it determines the switching time in liquid crystal displays. A special section is devoted to flow alignment and flow stability of nematic liquid crystals, which can be studied by obtaining the actual values of the viscosity coefficients or by direct shear flow simulations. The ever increasing speed of electronic computers has made it possible to study flow properties directly in very large systems. We finally review some work on the heat conductivity and the Lehmann effect, where a temperature gradient parallel to the cholesteric axis in a cholesteric liquid crystal causes the director to rotate. This rotation can also be induced by electric fields or flow of matter. This phenomenon and some of the flow properties are of technological importance for construction of molecular motors.