Extreme space weather events affect the stability and quality of the global navigation satellite systems (GNSS) of the second generation (GPS, GLONASS, Galileo, BeiDou/Compass) and GNSS augmentation. We review the theory about mechanisms behind the impact of geomagnetic storms, ionospheric irregularities, and powerful solar radio bursts on the GNSS user segment. We also summarize experimental observations of the space weather effects on GNSS performance in 2000–2020 to confirm the theory. We analyze the probability of failures in measurements of radio navigation parameters, decrease in positioning accuracy of GNSS users in dual-frequency mode and differential navigation mode (RTK), and in precise point positioning (PPP). Additionally, the review includes data on the occurrence of dangerous and extreme space weather phenomena and the possibility for predicting their im- pact on the GNSS user segment. The main conclusions of the review are as follows: 1) the positioning error in GNSS users may increase up to 10 times in various modes during extreme space weather events, as compared to the background level; 2) GNSS space and ground segments have been significantly modernized over the past decade, thus allowing a substantial in- crease in noise resistance of GNSS under powerful solar radio burst impacts; 3) there is a great possibility for increasing the tracking stability and accuracy of radio navigation parameters by introducing algorithms for adaptive lock loop tuning, taking into account the influence of space weather events; 4) at present, the urgent scientific and technical problem of modernizing GNSS by improving the scientific methodology, hardware and software for monitoring the system integrity and monitoring the availability of required navigation parameters, taking into account the impact of extreme space weather events, is still unresolved.