Sifting through vast textual data and summarizing key information imposes a substantial burden on how clinicians allocate their time. Although large language models (LLMs) have shown immense promise in natural language processing (NLP) tasks, their efficacy across diverse clinical summarization tasks has not yet been rigorously examined. In this work, we employ domain adaptation methods on eight LLMs, spanning six datasets and four distinct summarization tasks: radiology reports, patient questions, progress notes, and doctor-patient dialogue. Our thorough quantitative assessment reveals trade-offs between models and adaptation methods in addition to instances where recent advances in LLMs may not lead to improved results. Further, in a clinical reader study with six physicians, we depict that summaries from the best adapted LLM are preferable to human summaries in terms of completeness and correctness. Our ensuing qualitative analysis delineates mutual challenges faced by both LLMs and human experts. Lastly, we correlate traditional quantitative NLP metrics with reader study scores to enhance our understanding of how these metrics align with physician preferences. Our research marks the first evidence of LLMs outperforming human experts in clinical text summarization across multiple tasks. This implies that integrating LLMs into clinical workflows could alleviate documentation burden, empowering clinicians to focus more on personalized patient care and other irreplaceable human aspects of medicine.