The implementation of Intelligent Transport System (ITS) technology is expected to significantly improve road safety and traffic efficiency. One of the key components of ITS is precise vehicle positioning. Positioning with decimetre to sub-metre accuracy is a fundamental capability for self-driving, and other automated applications. Global Navigation Satellite System (GNSS) Precise Point Positioning (PPP) is an attractive positioning approach for ITS due to its relatively low-cost and flexibility. However, GNSS PPP is vulnerable to several effects, especially those caused by the challenging urban environments, where the ITS technology is most likely needed. To meet the high integrity requirements of ITS applications, it is necessary to carefully analyse potential faults and failures of PPP and to study relevant integrity monitoring methods. In this paper an overview of vulnerabilities of GNSS PPP is presented to identify the faults that need to be monitored when developing PPP integrity monitoring methods. These vulnerabilities are categorised into different groups according to their impact and error sources to assist integrity fault analysis, which is demonstrated with Failure Modes and Effects Analysis (FMEA) and Fault Tree Analysis (FTA) methods. The main vulnerabilities are discussed in detail, along with their causes, characteristics, impact on users, and related mitigation methods. In addition, research on integrity monitoring methods used for accounting for the threats and faults in PPP for ITS applications is briefly reviewed. Both system-level (network-end) and user-level (user-end) integrity monitoring approaches for PPP are briefly discussed, focusing on their development and the challenges in urban scenarios. Some open issues, on which further efforts should focus, are also identified.