Bayesian designs are found by maximising the expectation of a utility function where the utility function is chosen to represent the aim of the experiment. There are several hurdles to overcome when considering Bayesian design for intractable models. Firstly, common to nearly all Bayesian design problems, the expected utility function is not analytically tractable and requires approximation. Secondly, this approximate expected utility needs to be maximised over a potentially high-dimensional design space. To compound these problems, thirdly, the model is intractable, i.e. has no closed form. New approaches to maximise an approximation to the expected utility for intractable models are developed and applied to illustrative exemplar design problems with experimental aims of parameter estimation and model selection.