  We show that any isomorphism between mapping class groups of orientable
infinite-type surfaces is induced by a homeomorphism between the surfaces. Our
argument additionally applies to automorphisms between finite-index subgroups
of these `big' mapping class groups and shows that each finite-index subgroup
has finite outer automorphism group. As a key ingredient, we prove that all
simplicial automorphisms between curve complexes of infinite-type orientable
surfaces are induced by homeomorphisms.
