Fine-tuning large language models (LLMs) with limited data poses a practical challenge in low-resource languages, specialized domains, and constrained deployment settings. While pre-trained LLMs provide strong foundations, effective adaptation under data scarcity requires focused and efficient fine-tuning techniques. This paper presents a structured and practical survey of recent methods for fine-tuning LLMs in data-scarce scenarios. We systematically review parameter-efficient fine-tuning techniques that lower training and deployment costs, domain and cross-lingual adaptation methods for both encoder and decoder models, and model specialization strategies. We further examine preference alignment approaches that guide model behavior using limited human or synthetic feedback, emphasizing sample and compute efficiency. Throughout, we highlight empirical trade-offs, selection criteria, and best practices for choosing suitable techniques based on task constraints, including model scaling, data scaling, and the mitigation of catastrophic forgetting. The aim is to equip researchers and practitioners with actionable insights for effectively fine-tuning LLMs when data and resources are limited.