  In order to extrincate the structure of corings with a finitely generated and
projective generator we give the notion of a comatrix coring. As consequences
we give generalizations of the main characterizations of faithfully flat Galois
corings and extensions which work for corings without grouplike elements, as
well as a generalization of the Descent Theorem. We provide also a complete
description of all cosemisimple corings. No restrictions are made over the
ground noncommutative ring.
