This manuscript summarizes the new algorithm of Fast Precise Point Positioning (FPPP) the developed during the projects "Enhanced PPP GNSS multifrequency user algorithm" nd “Precise Real Time Orbit Determination and Time synchronisation”, both funded by the European Space Agency (ESA). The main innovations achieved during the overall project comprise the application of precise ionospheric corrections to facilitate the fast resolution of undifferenced carrier phase ambiguities, ambiguity validation and integrity monitoring for both multi- and single-frequency users. Among the integrity, detailed in previous works, the performance of the FPPP algorithm in terms of improved accuracy and convergence time is demonstrated with actual GPS and simulated Galileo data. The 10-centimeters error level real-time kinematic positioning can be achieved in few minutes for dual-and single-frequency users, almost instantaneous for three-frequency users (or once the tropospheric delay is well estimated in few minutes in cold start), and with very limited bandwidth requirements for the FPPP users (less than 300 bps for dual-frequency GPS).