  Moduli spaces of (polarized) Enriques surfaces can be described as open
subsets of modular varieties of orthogonal type. It was shown by Gritsenko and
Hulek that there are, up to isomorphism, only finitely many different moduli
spaces of polarized Enriques surfaces. Here we investigate the possible
arithmetic groups and show that there are exactly $87$ such groups up to
conjugacy. We also show that all moduli spaces are dominated by a moduli space
of polarized Enriques surfaces of degree $1240$. Ciliberto, Dedieu, Galati, and
Knutsen have also investigated moduli spaces of polarized Enriques surfaces in
detail. We discuss how our enumeration relates to theirs. We further compute
the Tits building of the groups in question. Our computation is based on groups
and indefinite quadratic forms and the algorithms used are explained.
