The attempt to utilize machine learning in PCG has been made in the past. In this survey paper, we investigate how generative artificial intelligence (AI), which saw a significant increase in interest in the mid-2010s, is being used for PCG. We review applications of generative AI for the creation of various types of content, including terrains, items, and even storylines. While generative AI is effective for PCG, one significant issues it faces is that building high-performance generative AI requires vast amounts of training data. Because content generally highly customized, domain-specific training data is scarce, and straightforward approaches to generative AI models may not work well. For PCG research to advance further, issues related to limited training data must be overcome. Thus, we also give special consideration to research that addresses the challenges posed by limited training data.