This paper concerns the analysis of how uncertainty propagates through large computational models like finite element models. If a model is expensive to run, a Monte Carlo approach based on sampling over the possible model inputs will not be feasible, because the large number of model runs will be prohibitively expensive. Fortunately, an alternative to Monte Carlo is available in the form of the established Bayesian algorithm discussed here; this algorithm can provide information about uncertainty with many less model runs than Monte Carlo requires. The algorithm also provides information regarding sensitivity to the inputs i.e. the extent to which input uncertainties are responsible for output uncertainty. After describing the basic principles of the Bayesian approach, it is illustrated via two case studies: the first concerns a finite element model of a human heart valve and the second, an airship model incorporating fluid structure interaction.