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Bayesian optimisation is a sample-efficient search methodology that holds great promise for accelerating drug and materials discovery programs. A frequently-overlooked modelling consideration in Bayesian optimisation strategies however, is the representation of heteroscedastic aleatoric uncertainty. In many practical applications it is desirable to identify inputs with low aleatoric noise, an example of which might be a material composition which consistently displays robust properties in response to a noisy fabrication process. In this paper, we propose a heteroscedastic Bayesian optimisation scheme capable of representing and minimising aleatoric noise across the input space. Our scheme employs a heteroscedastic Gaussian process (GP) surrogate model in conjunction with two straightforward adaptations of existing acquisition functions. First, we extend the augmented expected improvement (AEI) heuristic to the heteroscedastic setting and second, we introduce the aleatoric noise-penalised expected improvement (ANPEI) heuristic. Both methodologies are capable of penalising aleatoric noise in the suggestions and yield improved performance relative to homoscedastic Bayesian optimisation and random sampling on toy problems as well as on two real-world scientific datasets. Code is available at: \url{https://github.com/Ryan-Rhys/Heteroscedastic-BO}, Published in Machine Learning: Science and Technology 2021 (https://iopscience.iop.org/article/10.1088/2632-2153/ac298c) Earlier version accepted to the 2019 NeurIPS Workshop on Safety and Robustness in Decision Making