We evaluate four state-of-the-art instruction-tuned large language models (LLMs) -- ChatGPT, Flan-T5 UL2, Tk-Instruct, and Alpaca -- on a set of 13 real-world clinical and biomedical natural language processing (NLP) tasks in English, such as named-entity recognition (NER), question-answering (QA), relation extraction (RE), etc. Our overall results demonstrate that the evaluated LLMs begin to approach performance of state-of-the-art models in zero- and few-shot scenarios for most tasks, and particularly well for the QA task, even though they have never seen examples from these tasks before. However, we observed that the classification and RE tasks perform below what can be achieved with a specifically trained model for the medical field, such as PubMedBERT. Finally, we noted that no LLM outperforms all the others on all the studied tasks, with some models being better suited for certain tasks than others.