Abstract: Bounded cohomology is a functional analytic analogue of group cohomology, with many applications in rigidity theory, geometric group theory, and geometric topology. A major drawback is the lack of excision, and because of this some basic computations are currently out of reach; in particular the bounded cohomology of some ?small? groups, such as the free group, is still mysterious. On the other hand, in the past few years full computations have been carried out for some ?big? groups, most notably transformation groups of R^n, where the ordinary cohomology is not yet completely understood. I will report on this recent progress, which will include joint work with Caterina Campagnolo, Yash Lodha and Marco Moraschini, and joint work with Nicolas Monod, Sam Nariman and Alexander Kupers.