  We determine the bounded cohomology of the group of homeomorphisms of certain
low-dimensional manifolds. In particular, for the group of
orientation-preserving homeomorphisms of the circle and of the closed 2-disc,
it is isomorphic to the polynomial ring generated by the bounded Euler class.
These seem to be the first examples of groups for which the entire bounded
cohomology can be described without being trivial.
  We further prove that, contrary to ordinary cohomology, the diffeomorphisms
groups of the circle and of the closed 2-disc have the same bounded cohomology
as their homeomorphism groups and that both differ from the ordinary
cohomology.
  Finally, we determine the low-dimensional bounded cohomology of homeo- and
diffeomorphism of the spheres $S^n$ and of certain 3-manifolds. In particular,
we answer a question of Ghys by showing that the Euler class in
$H^4(Homeo_\circ(S^3))$ is unbounded.
