Abstract: Bounded cohomology for groups and spaces was originally defined by Gromov in the 80’s and it is intimately related to the geometric and dynamical properties of the groups. For example Ghys used the bounded Euler class to classify certain group actions on the circle up to (semi)conjugacy. However, unlike the group cohomology, it is notoriously difficult to calculate bounded cohomology of groups. And in fact there is no countably generated group known for which we can completely calculate the bounded cohomology unless it is trivial in all positive degrees like the case of amenable groups. In this talk, I will talk about joint work with Nicolas Monod on the bounded cohomology of certain homeomorphisms and diffeomorphism groups. In particular we show that the bounded cohomology of Diff(S^1) and Diff(D^2) are polynomial rings generated by the Euler class. If time permits, I also discuss our solution to Ghys’ question about generalizing Milnor-Wood inequality to flat S^3-bundles. In particular, we show that the Euler class for flat S^3-bundles is an unbounded class.