Clinical text summarization is vital for improving healthcare by providing concise insights from extensive medical data. Large Language Models (LLMs) demonstrate potential in automating this process, but their use comes with challenges like data privacy, adapting to specific medical contexts, and ensuring summaries are clinically relevant. This paper explores how models like GPT and ClinicalBERT can summarize electronic health records, discharge notes, and medical research papers. We discuss important datasets such as MIMIC-III, MIMIC-IV, i2b2, n2c2, MedMentions, and PubMed, which are essential for training LLMs in healthcare. By examining these datasets, we highlight their importance and the challenges they pose for summarization tasks. We also address issues like generalization across medical domains, maintaining factual accuracy, and the need for understandable machine-generated summaries. Finally, we outline potential future directions to improve LLMs in healthcare, including ethical considerations, refining datasets, and integrating explainable AI techniques.