World-class universities (WCUs), which are generally regarded as global research universities, are important constituents of the global higher education system. There are agreed-upon features of WCUs when compared with other research universities (RUs). However, there has been an absence of attempts to explore the special features of WCUs from a functional perspective that emphasizes a function which is unique to WCUs, apart from the widely-acknowledged three basic functions of education, research and service. Using a mixed research method, including the document analysis of 83 research universities, 74 semi-structured interviews with university leaders, academics and experts worldwide as well as an online survey for WCUs’ leaders and international academic experts (N = 118), this study examines the differences between WCUs and RUs in the three basic functions (that is, education, research and service), while at the same time investigating the special function or unique mission that is possessed exclusively by WCUs. This study identifies two core dimensions of WCUs’ special function or unique mission: to serve the global common good and to act as a global role model for research universities, with four main features: global positioning, global contribution, global influence and global cooperation. It then summarizes WCUs’ special function or unique mission in a word: Globalizing.