We develop a Bayesian framework for calibrating flood inundation simulators on an observation of flood extent, and making calibrated predictions of a future event. We illustrate the framework using the binary channel (BC) model for the likelihood of the observed flood extent given a simulation of flood extent. The BC model leads to poor results, and this motivates the search for a more appropriate likelihood model, which forms the basis for the rest of the thesis. We extend the Ising model to regression on a binary image and review methods for dealing with the intractable normalising constant. We propose novel applications of path sampling, extend path sampling to sampling over areas, and develop approximations to path sampling. We also develop the heterogeneous binary channel (HBC) model to test the effect of heterogeneity and spatial dependence. We extend the hidden conditional autoregressive (HCAR) model to regression on a binary image. We show that the limit of the HCAR model as the parameters approach the boundary is the (improper) hidden intrinsic autoregressive (HIAR) model. We prove that the HIAR model can be used for calibration but not calibrated prediction. We develop a number of methods for improving mixing of the MCMC algorithm. We explore two extensions of the HCAR model. First the heterogeneous HCAR (HHCAR) model, which represents heterogeneity, and second the continuous HCAR (CHCAR) model, which uses continuous simulation values. In conclusion, using our Bayesian framework we can replicate the results of less rigorous approaches, for example generalised likelihood uncertainty estimation (GLUE), and make probabilistic predictions which are not possible in these less rigorous approaches. Future work would further develop the likelihood models.