Large Language Models (LLMs) are increasingly utilised in software development, particularly for automatic code generation. Game development, a complex and multidisciplinary field, has begun leveraging LLMs to streamline various stages of the development process. This paper focuses on the application of LLMs in automating code generation for game development, providing a comprehensive survey of existing research from multiple perspectives. We analyse the benefits and limitations of LLM-based approaches in this domain and identify key challenges. Our findings highlight the potential of LLMs to enhance game development workflows, while outlining future research directions to address existing limitations.