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Deep learning has become the method of choice to tackle real-world problems in different domains, partly because of its ability to learn from data and achieve impressive performance on a wide range of applications. However, its success usually relies on two assumptions: (i) vast troves of labeled datasets are required for accurate model fitting, and (ii) training and testing data are independent and identically distributed. Its performance on unseen target domains, thus, is not guaranteed, especially when encountering out-of-distribution data at the adaptation stage. The performance drop on data in a target domain is a critical problem in deploying deep neural networks that are successfully trained on data in a source domain. Unsupervised domain adaptation (UDA) is proposed to counter this, by leveraging both labeled source domain data and unlabeled target domain data to carry out various tasks in the target domain. UDA has yielded promising results on natural image processing, video analysis, natural language processing, time-series data analysis, medical image analysis, etc. In this review, as a rapidly evolving topic, we provide a systematic comparison of its methods and applications. In addition, the connection of UDA with its closely related tasks, e.g., domain generalization and out-of-distribution detection, has also been discussed. Furthermore, deficiencies in current methods and possible promising directions are highlighted.