Machine learning (ML) on tabular data is ubiquitous, yet obtaining abundant high-quality tabular data for model training remains a significant obstacle. Numerous works have focused on tabular data augmentation (TDA) to enhance the original table with additional data, thereby improving downstream ML tasks. Recently, there has been a growing interest in leveraging the capabilities of generative AI for TDA. Therefore, we believe it is time to provide a comprehensive review of the progress and future prospects of TDA, with a particular emphasis on the trending generative AI. Specifically, we present an architectural view of the TDA pipeline, comprising three main procedures: pre-augmentation, augmentation, and post-augmentation. Pre-augmentation encompasses preparation tasks that facilitate subsequent TDA, including error handling, table annotation, table simplification, table representation, table indexing, table navigation, schema matching, and entity matching. Augmentation systematically analyzes current TDA methods, categorized into retrieval-based methods, which retrieve external data, and generation-based methods, which generate synthetic data. We further subdivide these methods based on the granularity of the augmentation process at the row, column, cell, and table levels. Post-augmentation focuses on the datasets, evaluation and optimization aspects of TDA. We also summarize current trends and future directions for TDA, highlighting promising opportunities in the era of generative AI. In addition, the accompanying papers and related resources are continuously updated and maintained in the GitHub repository at https://github.com/SuDIS-ZJU/awesome-tabular-data-augmentation to reflect ongoing advancements in the field.