Ubiquitous healthcare is becoming a reality thanks to the advances in sensing and communication technologies, which make it possible to provide monitoring and diagnosis services outside the premises of healthcare providers. The Internet of Things (IoT) is the main paradigm through which medical devices will be connected to the Internet, thereby empowering near-realtime health services and transforming a patient's physical space into a smart space. Recent developments in nanotechnology are giving rise to the Internet of Nano-Things, with a new set of finegrained and highly sophisticated healthcare applications that can be run inside the human body. In this paper, we outline a vision of the ubiquitous healthcare ecosystem and its architectural requirements in order to incorporate nanonetworks. We identify some of the envisioned IoNT healthcare applications and the IoNT requirements that are necessary to support the different application categories, as well as the underlying healthcare service opportunities. In order to understand the current status of implementation, we provide a brief analysis of the major efforts targeted at IoNT performance analysis and evaluation. We finally discuss the most pressing challenges that the IoNT paradigm poses for healthcare applications and services.