Deep learning has demonstrated remarkable performance across various tasks in medical imaging. However, these approaches primarily focus on supervised learning, assuming that the training and testing data are drawn from the same distribution. Unfortunately, this assumption may not always hold true in practice. To address these issues, unsupervised domain adaptation (UDA) techniques have been developed to transfer knowledge from a labeled domain to a related but unlabeled domain. In recent years, significant advancements have been made in UDA, resulting in a wide range of methodologies, including feature alignment, image translation, self-supervision, and disentangled representation methods, among others. In this paper, we provide a comprehensive literature review of recent deep UDA approaches in medical imaging from a technical perspective. Specifically, we categorize current UDA research in medical imaging into six groups and further divide them into finer subcategories based on the different tasks they perform. We also discuss the respective datasets used in the studies to assess the divergence between the different domains. Finally, we discuss emerging areas and provide insights and discussions on future research directions to conclude this survey.