Research on language technology for the development of medical applications is currently a hot topic in Natural Language Understanding and Generation. Thus, a number of large language models (LLMs) have recently been adapted to the medical domain, so that they can be used as a tool for mediating in human-AI interaction. While these LLMs display competitive performance on automated medical texts benchmarks, they have been pre-trained and evaluated with a focus on a single language (English mostly). This is particularly true of text-to-text models, which typically require large amounts of domain-specific pre-training data, often not easily accessible for many languages. In this paper, we address these shortcomings by compiling, to the best of our knowledge, the largest multilingual corpus for the medical domain in four languages, namely English, French, Italian and Spanish. This new corpus has been used to train Medical mT5, the first open-source text-to-text multilingual model for the medical domain. Additionally, we present two new evaluation benchmarks for all four languages with the aim of facilitating multilingual research in this domain. A comprehensive evaluation shows that Medical mT5 outperforms both encoders and similarly sized text-to-text models for the Spanish, French, and Italian benchmarks, while being competitive with current state-of-the-art LLMs in English.