This paper discusses telemedicine and the employment of advanced mobile technologies in smart healthcare delivery. It covers the technological advances in connected smart healthcare, including the roles of artificial intelligence, machine learning, 5G and IoT platforms, and other enabling technologies. It also presents the challenges and potential risks that could arise from delivering connected smart healthcare services. Healthcare delivery is witnessing revolutions engineered by the developments in mobile connectivity and the plethora of platforms, applications, sensors, devices, and equipment that go along with it. Human society is evolving fast in response to these technological developments, which are also pushing the connectivity-providing sector to create and adopt new waves of network technologies. Consequently, new communications technologies have been introduced into the healthcare system and many novel applications have been developed to make it easier for sharing data in various forms and volumes within health-related services. These applications have also made it possible for telemedicine to be effectively adopted. This paper provides an overview of some of the recent developments within the space of mobile connectivity and telemedicine.