This paper explores pun generation, which creatively transforms linguistic elements to create humor or evoke double meaning. Pun generation aims to maintain contextual relevance and consistency, and is useful in creative writing and entertainment. While much research on pun generation has been conducted in computational linguistics, a dedicated survey systematically examining this field has been lacking. Therefore, this paper comprehensively reviews pun generation datasets and methodologies, summarizes automated and human evaluation metrics, and suggests research challenges and future directions.