  Let M be a compact manifold, possibly with boundary. We show that the group
of homeomorphisms of M has the automatic continuity property: any homomorphism
from Homeo(M) to any separable group is necessarily continuous. This answers a
question of C. Rosendal. If N is a submanifold of M, the group of
homeomorphisms of M that preserve N also has this property. Various
applications of automatic continuity are discussed, including applications to
the topology and structure of groups of germs of homeomorphisms. In an appendix
with Frederic Le Roux we also show, using related techniques, that the group of
germs at a point of homeomorphisms of Rn is strongly uniformly simple.
