Antibodies are canonically Y-shaped multimeric proteins capable of highly\nspecific molecular recognition. The CDRH3 region located at the tip of variable\nchains of an antibody dominates antigen-binding specificity. Therefore, it is a\npriority to design optimal antigen-specific CDRH3 regions to develop\ntherapeutic antibodies. However, the combinatorial nature of CDRH3 sequence\nspace makes it impossible to search for an optimal binding sequence\nexhaustively and efficiently using computational approaches. Here, we present\n\\texttt{AntBO}: a combinatorial Bayesian optimisation framework enabling\nefficient \\textit{in silico} design of the CDRH3 region. Ideally, antibodies\nare expected to have high target specificity and developability. We introduce a\nCDRH3 trust region that restricts the search to sequences with favourable\ndevelopability scores to achieve this goal. For benchmarking, \\texttt{AntBO}\nuses the \\texttt{Absolut!} software suite as a black-box oracle to score the\ntarget specificity and affinity of designed antibodies \\textit{in silico} in an\nunconstrained fashion~\\citep{robert2021one}. The experiments performed for\n$159$ discretised antigens used in \\texttt{Absolut!} demonstrate the benefit of\n\\texttt{AntBO} in designing CDRH3 regions with diverse biophysical properties.\nIn under $200$ calls to black-box oracle, \\texttt{AntBO} can suggest antibody\nsequences that outperform the best binding sequence drawn from 6.9 million\nexperimentally obtained CDRH3s and a commonly used genetic algorithm baseline.\nAdditionally, \\texttt{AntBO} finds very-high affinity CDRH3 sequences in only\n38 protein designs whilst requiring no domain knowledge. We conclude\n\\texttt{AntBO} brings automated antibody design methods closer to what is\npractically viable for in vitro experimentation.\n