I will report on a recent work with Ron Douglas in which we investigated a question of Higson about cohomological characterization of exact groups. Exactness for groups is a weak, metric version of amenability and it turns out to be extremely useful in many settings such as index theory, isoperimetric problems, geometric group theory. I will discuss a vanishing theorem for bounded cohomology of exact groups in the spirit of a classical theorem of B.E.Johnson for amenable groups. Our main tool will be a characterization of exact groups in terms of the existence of invariant expectations, which are weak versions of invariant means.