The recent emergence of Large Language Models (LLMs) has enabled significant advances in the field of Natural Language Processing (NLP). While these new models have demonstrated superior performance on various tasks, their application and potential are still underexplored, both in terms of the diversity of tasks they can handle and their domain of application. In this context, we evaluate four state-of-the-art instruction-tuned LLMs (ChatGPT, Flan-T5 UL2, Tk-Instruct, and Alpaca) on a set of 13 real-world clinical and biomedical NLP tasks in English, including named-entity recognition (NER), question-answering (QA), relation extraction (RE), and more. Our overall results show that these evaluated LLMs approach the performance of state-of-the-art models in zero- and few-shot scenarios for most tasks, particularly excelling in the QA task, even though they have never encountered examples from these tasks before. However, we also observe that the classification and RE tasks fall short of the performance achievable with specifically trained models designed for the medical field, such as PubMedBERT. Finally, we note that no single LLM outperforms all others across all studied tasks, with some models proving more suitable for certain tasks than others.