Pun generation seeks to creatively modify linguistic elements in text to produce humour or evoke double meanings. It also aims to preserve coherence and contextual appropriateness, making it useful in creative writing and entertainment across various media and contexts. This field has been widely studied in computational linguistics, while there are currently no surveys that specifically focus on pun generation. To bridge this gap, this paper provides a comprehensive review of pun generation datasets and methods across different stages, including traditional approaches, deep learning techniques, and pre-trained language models. Additionally, we summarise both automated and human evaluation metrics used to assess the quality of pun generation. Finally, we discuss the research challenges and propose promising directions for future work.