  We study the bounded cohomology of certain groups acting on the Cantor set.
More specifically, we consider the full group of homeomorphisms of the Cantor
set as well as Thompson's group $V$. We prove that both of these groups are
boundedly acyclic, that is the bounded cohomology with trivial real
coefficients vanishes in positive degrees. Combining this result with the
already established $\mathbb{Z}$-acyclicity of Thompson's group $V$, will make
$V$ the first example of a finitely generated group, in fact the first example
of a group of type $F_\infty$, which is universally boundedly acyclic. Before
proving bounded acyclicity, we gather various properties of the groups under
consideration and certain subgroups thereof. As a consequence the proofs of
bounded acyclicity will be relatively short. It will turn out that the
approaches to handle these groups are very similar. This suggests that there
could be a unifying approach which would imply the bounded acyclicity of a
larger class of groups acting on the Cantor set, including the discussed ones.
