We analyse the behaviour of approximate Bayesian computation (ABC) when the model generating the simulated data differs from the actual data-generating process, i.e. when the data simulator in ABC is misspecified. We demonstrate both theoretically and in simple, but practically relevant, examples that when the model is misspecified different versions of ABC can yield substantially different results. Our theoretical results demonstrate that even though the model is misspecified, under regularity conditions, the accept–reject ABC approach concentrates posterior mass on an appropriately defined pseudotrue parameter value. However, under model misspecification the ABC posterior does not yield credible sets with valid frequentist coverage and has non-standard asymptotic behaviour. In addition, we examine the theoretical behaviour of the popular local regression adjustment to ABC under model misspecification and demonstrate that this approach concentrates posterior mass on a pseudotrue value that is completely different from accept–reject ABC. Using our theoretical results, we suggest two approaches to diagnose model misspecification in ABC. All theoretical results and diagnostics are illustrated in a simple running example.