Launched on 22 November 2013 by the European Space Agency (ESA), the three Swarm satellites were initially designed with their original configuration to monitor and understand the geomagnetic field and the state of the ionosphere and magnetosphere. In 2017, for the first time, some pre- and post-earthquake magnetic field anomalies as recorded by Swarm satellites were revealed on occasion of the 2015 Nepal M7.8 earthquake. Interestingly, the cumulative number of satellite anomalies behaved as the cumulative number of earthquakes, with the so-called S-shape, providing a heuristic proof on the lithospheric origin of the satellite anomalies (De Santis et al., 2017; https://doi.org/10.1016/j.epsl.2016.12.037). Following the same approach, other promising results were obtained for 12 case studies in a range of earthquake magnitude 6.1-8.3, investigated with the support of ESA to INGV (and Planetek) funding the SAFE (SwArm For Earthquake study) project (De Santis et al., 2019a; https://doi.org/10.3390/atmos10070371). In 2019, almost five years of Swarm magnetic field and electron density data were analysed with a Superposed Epoch and Space approach and correlated with major worldwide M5.5+ earthquakes (De Santis et al. 2019b; https://doi.org/10.1038/s41598-019-56599-1). The analysis confirmed the correlation between satellite anomalies and earthquakes above any reasonable doubt, by means of a statistical comparison with random simulations of anomalies. It also confirmed the Rikitake (1987) law, initially proposed for ground data: the larger the magnitude of the impending earthquake, the longer the precursory time of anomaly appearance in ionosphere from satellite. Furthermore, we demonstrated in several case studies (e.g. Akhoondzadeh et al. 2019; https://doi.org/10.1016/j.asr.2019.03.020; De Santis et al. 2020; https://doi.org/10.3389/feart.2020.540398) that the integration of Swarm data with other kinds of measurements from ground, atmosphere and space (e.g. CSES data) reveals a chain of processes before mainshocks of many seismic sequences. A review of the above results together with some new ones will be presented.