  We show that topological amenability of an action of a countable discrete
group on a compact space is equivalent to the existence of an invariant mean
for the action. We prove also that this is equivalent to vanishing of bounded
cohomology for a class of Banach G-modules associated to the action, as well as
to vanishing of a specific cohomology class. In the case when the compact space
is a point our result reduces to a classic theorem of B.E. Johnson
characterising amenability of groups. In the case when the compact space is the
Stone-\v{C}ech compactification of the group we obtain a cohomological
characterisation of exactness for the group, answering a question of Higson.
