Whereas the use of traditional Monte Carlo simulation requires probability distributions for the uncertain parameters entering the system, distributionally robust Monte Carlo simulation does not. According to this new theory, instead of carrying out simulations using some rather arbitrary probability distribution such as Gaussian for the uncertain parameters, we provide a rather different prescription based on distributional robustness considerations. Motivated by manufacturing considerations, a class of distributions F is specified and the results of the simulation hold for all f in F. This new method of Monte Carlo simulation was developed with the robustician in mind in that we begin only with bounds on the uncertain parameters and no a priori probability distribution is assumed.