Recent years have seen an explosive increase in research on large language models (LLMs), and accompanying public engagement on the topic. While starting as a niche area within natural language processing, LLMs have shown remarkable potential across a broad range of applications and domains, including games. This paper surveys the current state of the art across the various applications of LLMs in and for games, and identifies the different roles LLMs can take within a game. Importantly, we discuss underexplored areas and promising directions for future uses of LLMs in games and we reconcile the potential and limitations of LLMs within the games domain. As the first comprehensive survey and roadmap at the intersection of LLMs and games, we are hopeful that this paper will serve as the basis for groundbreaking research and innovation in this exciting new field.