  We give necessary and sufficient criteria for a smooth Enriques surface S in
P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove
in many cases that, when S contains plane cubic curves, the intersection of the
quadrics containing S is the union of S and the 2-planes spanned by the plane
cubic curves. We also give a new (very quick) proof of the projective normality
of S if the degree of S is at least 12.
