  We study injective homomorphisms between big mapping class groups of
infinite-type surfaces. First, we construct (uncountably many) examples of
surfaces without boundary whose (pure) mapping class groups are not co-Hopfian;
these are the first examples of injective endomorphisms of mapping class groups
(of surfaces with empty boundary) that fail to be surjective.
  We then prove that, subject to some topological conditions on the domain
surface, any continuous injective homomorphism between (arbitrary) big mapping
class groups that sends Dehn twists to Dehn twists is induced by homeomorphism.
  Finally, we explore the extent to which, in stark contrast to the finite-type
case, superinjective maps between curve graphs impose no topological
restrictions on the underlying surfaces.
