Abstract: In 1896 Enriques constructed an example of a smooth
non-rational surface with invariants q = p_g = 0. It arises as the
minimal desingularization of an Enriques sextic, i.e a sextic surface
that is non-normal along the edges of a tetrahedron. In modern
terminology, these are examples of Enriques surfaces, and it was
already known to the italian geometers that a general complex Enriques
surface arises from this construction. I will talk about a joint work
with G. Martin and D. Veniani, in which we show that in fact every
Enriques surface (in characteristic different from 2) arises as the
minimal desingularization of an Enriques sextic. In a similar vein, we
show that every Enriques surface containing a smooth rational curve
arises as a certain congruence of lines in projective 3-space, called
a Reye congruence.