  Euclid will observe 15 000 deg$^2$ of the darkest sky, in regions free of
contamination by light from our Galaxy and our Solar System. Three "Euclid Deep
Fields" surveys covering around 40 deg$^2$ in total will extend the scientific
scope of the mission to the high-redshift Universe. The complete survey will be
constituted by hundreds of thousands of images and several tens of petabytes of
data. About 10 billion sources will be observed. With these images Euclid will
probe the expansion history of the Universe and the evolution of cosmic
structures. This will be achieved by measuring the effect on galaxy shapes due
to dark matter gravitational lensing, and by reconstructing the
three-dimensional distribution of cosmic structures from the measured
spectroscopic redshifts of galaxies and clusters of galaxies. These proceedings
present the implications for cosmology and cosmological constraints of this
unprecedented data set. Of particular interest are the expected constraints on
the nature of dark energy.
