Bayesian optimization has become widely popular across various experimental sciences due to its favorable attributes: it can handle noisy data, perform well with relatively small datasets, and provide adaptive suggestions for sequential experimentation. While still in its infancy, Bayesian optimization has recently gained traction in bioprocess engineering. However, experimentation with biological systems is highly complex and the resulting experimental uncertainty requires specific extensions to classical Bayesian optimization. Moreover, current literature often targets readers with a strong statistical background, limiting its accessibility for practitioners. In light of these developments, this review has two aims: first, to provide an intuitive and practical introduction to Bayesian optimization; and second, to outline promising application areas and open algorithmic challenges, thereby highlighting opportunities for future research in machine learning.