The detection of more than 300 extrasolar planets orbiting other solar-like stars opened the window to a new field of astrophysics. Many projects to search for Earth-like planets are currently under way, using a huge battery of telescopes and instruments. New instrumentation is also being developed towards this goal for use in both ground- and space-based based facilities. Since planets come as an output of the star formation process, the study of the stars hosting planets is of great importance. The stellar-planet connection is strengthened by the fact that most of the exoplanets were discovered using a Doppler radial-velocity technique, where the gravitational influence of the planet on the star and not the planet itself is actually measured. This project aims at doing frontier research to explore i) in unique detail the stellar limitations of the radial-velocity technique, as well as ways of reducing them, having in mind the detection of Earth-like planets and ii) to develop and apply software packages aiming at the study of the properties of the planet-host stars, having in mind the full characterization of the newfound planets, as well as understanding planet formation processes. These goals will improve our capacity to detect, study, and characterize new very low mass extra-solar planets. EXOEarths further fits into the fact that I am currently Co-PI of the project for a new high-resolution ultra-stable spectrograph for the VLT. The results of this project are crucial to fully exploit this new instrument. They will be also of extreme importance to current state-of-the-art planet-search projects aiming at the discovery of other Earths, in particular those making use of the radial-velocity method.