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Open-set domain adaptation is a developing and practical solution to training deep networks using unlabeled data which have been collected among unknown data and are under domain shift with other labeled data. This scenario transfers knowledge from a source domain enriched with labeled data to the unlabeled target domain, meanwhile, unknown target samples which are not present in the source domain are separated. Existing methods aim to bridge the domain gap of shared classes between source and target domains in a trustworthy manner and avoid negative transfer learning using keeping away unknown data from the domain alignment step. In this review article, we present a unified framework of theory advances for network risk and a new categorization of open-set domain adaptation along with listing evaluation metrics and popular datasets. Then we accentuate challenges and gaps in existing studies to organize a road map for future research using detailed analysis of investigations. To bring things full circle, we also point out different assumptions and outlooks in the settings of this research area. • For the first time, existing methods in deep domain adaptation with open-set target data are categorized and reviewed. • To produce an optimum upper bound for network risk in the training step, needed theories are presented. • For each category of open-set domain adaptation methods, the problem assumptions along with their algorithms are pointed out. • To present a comparative framework for these methods, their performances, advantages and disadvantages are also examined. • Finally, for paving the research directions in future works, research gaps and challenges are discussed. [ABSTRACT FROM AUTHOR]