The Euclid mission, part of ESA's Cosmic Vision 2015-2025 program, aims to map the geometry of the dark universe by studying dark energy and dark matter. The mission, scheduled for launch in 2018, will use weak gravitational lensing and baryonic acoustic oscillations to investigate these unknown components of the universe. The Phase A study, completed in July 2011, outlined the mission's design and key scientific objectives. Euclid will survey 15,000 square degrees of the extragalactic sky, measuring the shapes and redshifts of galaxies to study dark matter and dark energy. The mission will also provide insights into the early universe, testing inflation models and constraints on neutrino masses. The payload includes a 1.2-meter telescope with visible and near-infrared instruments, capable of high-resolution imaging and spectroscopy. The mission is expected to have a significant impact on cosmology and astrophysics, with legacy data spanning decades.The Euclid mission, part of ESA's Cosmic Vision 2015-2025 program, aims to map the geometry of the dark universe by studying dark energy and dark matter. The mission, scheduled for launch in 2018, will use weak gravitational lensing and baryonic acoustic oscillations to investigate these unknown components of the universe. The Phase A study, completed in July 2011, outlined the mission's design and key scientific objectives. Euclid will survey 15,000 square degrees of the extragalactic sky, measuring the shapes and redshifts of galaxies to study dark matter and dark energy. The mission will also provide insights into the early universe, testing inflation models and constraints on neutrino masses. The payload includes a 1.2-meter telescope with visible and near-infrared instruments, capable of high-resolution imaging and spectroscopy. The mission is expected to have a significant impact on cosmology and astrophysics, with legacy data spanning decades.