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Bayesian inference is a powerful tool in gravitational-wave astronomy. Itenables us to deduce the properties of merging compact-object binaries and todetermine how these mergers are distributed as a population according to mass,spin, and redshift. As key results are increasingly derived using Bayesianinference, there is increasing scrutiny on Bayesian methods. In this review, wediscuss the phenomenon of \textit{model misspecification}, in which resultsobtained with Bayesian inference are misleading because of deficiencies in theassumed model(s). Such deficiencies can impede our inferences of the trueparameters describing physical systems. They can also reduce our ability todistinguish the "best fitting" model: it can be misleading to say that Model~Ais preferred over Model~B if both models are manifestly poor descriptions ofreality. Broadly speaking, there are two ways in which models fail: models thatfail to adequately describe the data (either the signal or the noise) havemisspecified likelihoods. Population models -- designed, for example, todescribe the distribution of black hole masses -- may fail to adequatelydescribe the true population due to a misspecified prior. We recommend testsand checks that are useful for spotting misspecified models using examplesinspired by gravitational-wave astronomy. We include companion python notebooksto illustrate essential concepts.

When models fail: an introduction to posterior predictive checks and model misspecification in gravitational-wave astronomy