Artificial Intelligence (AI) and its applications in modern industry, particularly in additive manufacturing (AM), have garnered significant attention from both researchers and industry professionals. In recent years, the rise of generative AI (GenAI) models, marked by the launch of ChatGPT in 2022, has driven an unprecedented shift in industry practices and workforce dynamics. This paper investigates the current state of research at the intersection of GenAI and AM, while also exploring future directions. By conducting a keyword search across four major databases (Web of Science, Scopus, IEEE Xplore, and ScienceDirect), we collected 272 papers. Through an in-depth evaluation of these papers, we identified five primary themes shaping GenAI applications in AM: AM process optimization, generative design, data & quality, design workflow, and applications. This paper also provides a detailed analysis of each category, offering valuable insights into the evolving role of GenAI in AM.