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Bayesian optimisation is a statistical method that efficiently models and optimises expensive “black-box” functions. This review considers the application of Bayesian optimisation to experimental design, in comparison to existing Design of Experiments (DOE) methods. Solutions are surveyed for a range of core issues in experimental design including: the incorporation of prior knowledge, high dimensional optimisation, constraints, batch evaluation, multiple objectives, multi-fidelity data, and mixed variable types.