Unsupervised domain adaptation (UDA) via deep learning has attracted appealing attention for tackling domain-shift problems caused by distribution discrepancy across different domains. Existing UDA approaches highly depend on the accessibility of source domain data, which is usually limited in practical scenarios due to privacy protection, data storage and transmission cost, and computation burden. To tackle this issue, many source-free unsupervised domain adaptation (SFUDA) methods have been proposed recently, which perform knowledge transfer from a pre-trained source model to unlabeled target domain with source data inaccessible. A comprehensive review of these works on SFUDA is of great significance. In this paper, we provide a timely and systematic literature review of existing SFUDA approaches from a technical perspective. Specifically, we categorize current SFUDA studies into two groups, i.e., white-box SFUDA and black-box SFUDA, and further divide them into finer subcategories based on different learning strategies they use. We also investigate the challenges of methods in each subcategory, discuss the advantages/disadvantages of white-box and black-box SFUDA methods, conclude the commonly used benchmark datasets, and summarize the popular techniques for improved generalizability of models learned without using source data. We finally discuss several promising future directions in this field.