  In this paper we consider moduli spaces of polarized and numerically
polarized Enriques surfaces. The moduli spaces of numerically polarized
Enriques surfaces can be described as open subsets of orthogonal modular
varieties of dimension 10. One of the consequences of our description is that
there are only finitely many isomorphism classes of moduli spaces of polarized
and numerically polarized Enriques surfaces. We use modular forms to prove for
a number of small degrees that the Kodaira dimension of the moduli space of
numerically polarized Enriques surfaces is negative. Finally we prove that
there are infinitely many polarizatons for which the moduli space of
numerically polarized Enriques surfaces is birational to the moduli space of
unpolarized Enriques surfaces with a level 2 structure.
