In recent years, we have witnessed dramatic developments of mobile healthcare robots, which enjoy many advantages over their human counterparts. Previous communication networks for healthcare robots always suffer from high response latency and/or time-consuming computing demands. Robust and high-speed communications and swift processing are critical, sometimes vital in particular in the case of healthcare robots, to the healthcare receivers. As a promising solution, offloading delay-sensitive and communicating-intensive tasks to the robot is expected to improve the services and benefit users. In this paper, we review several state-of-the-art technologies, such as the human-robot interface, environment and user status perceiving, navigation, robust communication and artificial intelligence, of a mobile healthcare robot and discuss in details the customized demands over offloading the computation and communication tasks. According to the intrinsic demands of tasks over the network usage, we categorize abilities of a typical healthcare robot into alternative classes: the edge functionalities and the core functionalities. Many latency-sensitive tasks, such as user interaction, or time-consuming tasks including health receiver status recognition and autonomous moving, can be processed by the robot without frequent communications with data centers. On the other hand, several fundamental abilities, such as radio resource management, mobility management, service provisioning management, need to update the main body with the cutting-edge artificial intelligence. Robustness and safety, in this case, are the primary goals in wireless communications that AI may provide ground-breaking solutions. Based on this partition, this article refers to several state-of-the-art technologies of a mobile healthcare robot and reviews some challenges to be met for its wireless communications.