Vehicular networks allow billions of vehicular users to be connected to report and exchange real-time data for offering various services, such as navigation, ride-hailing, smart parking, traffic monitoring, and vehicular digital forensics. Fifth generation (5G) is a new radio access technology with greater coverage, accessibility, and higher network density. 5G-supported Vehicular Networks (5GVNs) have attracted plenty of attention from both academia and industry. Geared with new features, they are expected to revolutionize the mobility ecosystem to empower a portfolio of new services. Meanwhile, the development of such communication capabilities, along with the development of sensory devices and the enhancement of local computing powers, have lead to an inevitable reality of massive data (e.g., identity, location, and trajectory) collection from vehicular users. Unfortunately, 5GVN are still confronted with a variety of privacy threats. Such threats are targeted at users’ data, identity, location, and trajectory. If not properly handled, such threats will cause unimaginable consequences to users. In this survey, we first review the state-of-the-art of survey papers. Next, we introduce the architecture, features, and services of 5GVN, followed by the privacy objectives of 5GVN and privacy threats to 5GVN. Further, we present existing privacy-preserving solutions and analyze them in-depth. Finally, we define some future research directions to draw more attention and down-to-earth efforts into this new architecture and its privacy issues.