Abstract: Bayesian optimisation (BO) is a powerful framework for global optimisation of costly functions, using predictions from Gaussian process models (GPs). In this work, we apply BO to functions that exhibit *invariance* to a known group of transformations. We show that vanilla BO algorithms are inefficient when optimising such invariant objectives, and provide a method for incorporating group invariances into the kernel of the GP to produce *invariance-aware* algorithms that achieve significant improvements in sample efficiency. In particular, we derive a bound on the maximum information gain of these invariant kernels, and provide novel upper and lower bounds on the number of observations required for invariance-aware BO algorithms to achieve $\epsilon$-optimality. We demonstrate our method's improved performance on a range of synthetic functions. We also apply our method to a *partially specified* setting, where the kernel is only invariant to a subset of the group, and find that these kernels achieve similar gains in sample efficiency at significantly reduced computational cost. Finally, we use invariant BO to design a current drive system for a nuclear fusion reactor, finding a high-performance solution where non-invariant methods failed.