Title: Antifragile and Robust Heteroscedastic Bayesian Optimisation. Abstract:     Bayesian Optimisation is an important decision-making tool for high-stakes applications in drug discovery and materials design. An oft-overlooked modelling consideration however is the representation of input-dependent or heteroscedastic aleatoric uncertainty. The cost of misrepresenting this uncertainty as being homoscedastic could be high in drug discovery applications where neglecting heteroscedasticity in high throughput virtual screening could lead to a failed drug discovery program. In this paper, we propose a heteroscedastic Bayesian Optimisation scheme which both represents and optimises aleatoric noise in the suggestions. We consider cases such as drug discovery where we would like to minimise or be robust to aleatoric uncertainty but also applications such as materials discovery where it may be beneficial to maximise or be antifragile to aleatoric uncertainty. Our scheme features a heteroscedastic Gaussian Process (GP) as the surrogate model in conjunction with two acquisition heuristics. First, we extend the augmented expected improvement (AEI) heuristic to the heteroscedastic setting and second, we introduce a new acquisition function, aleatoric-penalised expected improvement (ANPEI) based on a simple scalarisation of the performance and noise objective. Both methods are capable of penalising or promoting aleatoric noise in the suggestions and yield improved performance relative to a naive implementation of homoscedastic Bayesian Optimisation on toy problems as well as a real-world optimisation problem.