Context. The ESA Euclid mission has been designed to map the geometry of the dark Universe. Scheduled for launch in 2020, it will conduct a six-year visible and near-infrared imaging and spectroscopic survey over 15 000 deg² down to V_AB ~ 24.5. Although the survey will avoid ecliptic latitudes below 15°, the survey pattern in repeated sequences of four broadband filters seems well-adapted to detect and characterize solar system objects (SSOs). Aims. We aim at evaluating the capability of Euclid of discovering SSOs and of measuring their position, apparent magnitude, and spectral energy distribution. We also investigate how the SSO orbits, morphology (activity and multiplicity), physical properties (rotation period, spin orientation, and 3D shape), and surface composition can be determined based on these measurements. Methods. We used the current census of SSOs to extrapolate the total amount of SSOs that will be detectable by Euclid, that is, objects within the survey area and brighter than the limiting magnitude. For each different population of SSO, from neighboring near-Earth asteroids to distant Kuiper-belt objects (KBOs) and including comets, we compared the expected Euclid astrometry, photometry, and spectroscopy with the SSO properties to estimate how Euclid will constrain the SSOs dynamical, physical, and compositional properties. Results. With the current survey design, about 150 000 SSOs, mainly from the asteroid main-belt, should be observable by Euclid. These objects will all have high inclination, which is a difference to many SSO surveys that focus on the ecliptic plane. Euclid may be able to discover several 10⁴ SSOs, in particular, distant KBOs at high declination. The Euclid observations will consist of a suite of four sequences of four measurements and will refine the spectral classification of SSOs by extending the spectral coverage provided by Gaia and the LSST, for instance, to 2 microns. Combined with sparse photometry such as measured by Gaia and the LSST, the time-resolved photometry will contribute to determining the SSO rotation period, spin orientation, and 3D shape model. The sharp and stable point-spread function of Euclid will also allow us to resolve binary systems in the Kuiper belt and detect activity around Centaurs. Conclusions. The depth of the Euclid survey (V_AB ~ 24.5), its spectral coverage (0.5 to 2.0 μm), and its observation cadence has great potential for solar system research. A dedicated processing for SSOs is being set up within the Euclid consortium to produce astrometry catalogs, multicolor and time-resolved photometry, and spectral classification of some 10⁵ SSOs, which will be delivered as Legacy Science.