For a variety of crucial applications, Global Navigation Satellite System (GNSS) users not only require high accuracy positioning results, but also high reliability. Integrity monitoring can be used to assure secure and trustworthy positioning results under extreme risk probabilities. This report will talk about the key technique challenges of GNSS integrity monitoring from the geodetic background, including the DIA theory and robust estimation used for fault detection and exclusion purpose, as well as some key problems on the current PL calculation algorithms. Some recent research progresses on this topic will be presented, which specifically focuses on constructing a reliable stochastic model for PPP/PPP-RTK integrity monitoring by considering the atmosphere-related (ionospheric delays and tropospheric delays) and receiver-related error terms.