The fifth generation (5G) of communication systems has ambitious targets of data rate, end-to-end latency, and connection availability, while the deployment of a new flexible network architecture will spur new applications. E-health and mobile health (m-health) solutions will meet the increasing demand of new, sustainable, and more accessible services beneficial to both practitioners and the rapidly aging population. This paper aims at defining the technical requirements of future cellular networks to support a variety of advanced healthcare services (e.g., smart hospital, telesurgery, connected ambulances, and monitoring). While 5G will be able to satisfy these requirements, it will also pave the way for future e- and m-health in the sixth-generation (6G) cellular networks.