  The third author has shown that Shelah's eventual categoricity conjecture
holds in universal classes: class of structures closed under isomorphisms,
substructures, and unions of chains. We extend this result to the framework of
multiuniversal classes. Roughly speaking, these are classes with a closure
operator that is essentially algebraic closure (instead of, in the universal
case, being essentially definable closure). Along the way, we prove in
particular that Galois (orbital) types in multiuniversal classes are determined
by their finite restrictions, generalizing a result of the second author.
