  A group is boundedly acyclic if its bounded cohomology with trivial real
coefficients vanishes in all positive degrees. Amenable groups are boundedly
acyclic, while the first non-amenable examples were the group of compactly
supported homeomorphisms of $\mathbb{R}^n$ (Matsumoto--Morita) and mitotic
groups (L\"oh). We prove that binate (alias pseudo-mitotic) groups are
boundedly acyclic, which provides a unifying approach to the aforementioned
results. Moreover, we show that binate groups are universally boundedly
acyclic.
  We obtain several new examples of boundedly acyclic groups as well as
computations of the bounded cohomology of certain groups acting on the circle.
In particular, we discuss how these results suggest that the bounded cohomology
of the Thompson groups $F$, $T$, and $V$ is as simple as possible.
