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Antibodies are multimeric proteins capable of highly specific molecular recognition. The complementarity determining region 3 of the antibody variable heavy chain (CDRH3) often dominates antigen-binding specificity. Hence, it is a priority to design optimal antigen-specific CDRH3 to develop therapeutic antibodies. The combinatorial structure of CDRH3 sequences makes it impossible to query binding-affinity oracles exhaustively. Moreover, antibodies are expected to have high target specificity and developability. Here, we present AntBO, a combinatorial Bayesian optimization framework utilizing a CDRH3 trust region for an in silico design of antibodies with favorable developability scores. The in silico experiments on 159 antigens demonstrate that AntBO is a step toward practically viable in vitro antibody design. In under 200 calls to the oracle, AntBO suggests antibodies outperforming the best binding sequence from 6.9 million experimentally obtained CDRH3s. Additionally, AntBO finds very-high-affinity CDRH3 in only 38 protein designs while requiring no domain knowledge., Competing Interests: This article is an open-source research contribution by Huawei, Tech R&D (UK). We release all used resources on GitHub. This work was carried out while A.K. was previously employed as a research scientist intern position, and A.I.C.-R. was previously employed a research scientist position at Huawei, Tech R&D (UK), and Huawei owns all intellectual property rights in the work detailed herein. A.G., D.-G.-X.D., R.T., J.W., and H.B.-A. are currently affiliated with Huawei. V.G. holds advisory board positions in aiNET GmbH and Enpicorm B.V. and is also a consultant for Roche/Genetech., (© 2022 The Authors.)