Robotic telesurgery has a potential to provide extreme and urgent health care services and bring unprecedented opportunities to deliver highly specialized skills globally. It has a significant societal impact and is regarded as one of the appealing use cases of Tactile Internet and 5G applications. However, the performance of robotic telesurgery largely depends on the network performance in terms of latency, jitter and packet loss, especially when telesurgical system is equipped with haptic feedback. This imposes significant challenges to design a reliable and secure but cost-effective communication solution. This article aims to give a better understanding of the characteristics of robotic telesurgical system, and the limiting factors, the possible telesurgery services and the communication quality of service (QoS) requirements of the multi-modal sensory data. Based on this, a viable network architecture enabled by the converged edge and core cloud is presented and the relevant research challenges, open issues and enabling technologies in the 5G communication system are discussed.