Methods for story generation with Large Language Models (LLMs) have come into the spotlight recently. We create a novel taxonomy of LLMs for story generation consisting of two major paradigms: (i) independent story generation by an LLM, and (ii) author-assistance for story generation – a collaborative approach with LLMs supporting human authors. We compare existing works based on their methodology, datasets, generated story types, evaluation methods, and LLM usage. With a comprehensive survey, we identify potential directions for future work