Fine-tuning is a common approach for large language models (LLMs) to acquire domain-specific knowledge for downstream tasks. However, datasets used in this process often contain sensitive information, raising privacy concerns due to the potential for data leakage during unprotected training. Differential Privacy (DP), a widely adopted privacy-preserving technique, addresses this issue by adding carefully calibrated noise to gradients or parameter updates, thereby quantifying and controlling information leakage during training. In this survey, we systematically review existing methodologies for applying differential privacy in the fine-tuning of large language models. We analyze these methods regarding their effectiveness in protecting privacy and their impact on model performance. Furthermore, we highlight the limitations observed in current approaches and identify critical research gaps, providing a clear overview of ongoing challenges and promising future directions for improving differentially private fine-tuning of LLMs.