Log data, generated by software systems, provides crucial insights for tasks like monitoring, root cause analysis, and anomaly detection. Due to the vast volume of logs, automated log parsing is essential to transform semi-structured log messages into structured representations. Traditional log parsing techniques often require manual configurations, such as defining log formats or labeling data, which limits scalability and usability. Recent advances in large language models (LLMs) have introduced the new research field of LLM-based log parsing, offering potential improvements in automation and adaptability. Despite promising results, there is no structured overview of these approaches since this is a relatively new research field with the earliest advances published in late 2023. This paper systematically reviews 29 LLM-based log parsing methods, comparing their capabilities, limitations, and reliance on manual effort. We analyze the learning and prompt-engineering paradigms employed, efficiency- and effectiveness-enhancing techniques, and the role of LLMs in the parsing process. We aggregate the results of the survey in a large table comprising the characterizing features of LLM-based log parsing approaches and derive the general process of LLM-based log parsing, incorporating all reviewed approaches in a single flow chart. Additionally, we benchmark seven open-source LLM-based log parsers on public datasets and critically assess their reproducibility. Our findings summarize the advances of this new research field and provide insights for researchers and practitioners seeking efficient and user-friendly log parsing solutions, with all code and results made publicly available for transparency.