Given the overwhelming impact of machine learning on the last decade, several libraries and frameworks have been developed in recent years to simplify the design and training of neural networks, providing array-based programming, automatic differentiation and user-friendly access to hardware accelerators. None of those tools, however, was designed with native and transparent support for Cloud Computing or heterogeneous High-Performance Computing (HPC). The DeepHealth Toolkit is an open source deep learning toolkit aimed at boosting productivity of data scientists operating in the medical field by providing a unified framework for the distributed training of neural networks, that is able to leverage hybrid HPC and Cloud environments in a way transparent to the user. The toolkit is composed of a computer vision library, a deep learning library, and a front-end for non-expert users; all of the components are focused on the medical domain, but they are general purpose and can be applied to any other field. In this paper, the principles driving the design of the DeepHealth libraries are described, along with details about the implementation and the interaction between the different elements composing the toolkit. Finally, experiments on common benchmarks prove the efficiency of each separate component, and of the DeepHealth Toolkit overall.