Bounded cohomology is a functional-analytic variant of ordinary cohomology with applications in the study of manifold geometry, rigidity theory, ergodic theory, and many other areas. The aim of this talk is to discuss a result concerning the vanishing of the bounded cohomology of the transformation groups of Euclidean spaces. To this end, we will first recall the definition of (bounded) cohomology of groups, along with its main properties and key results. We will then briefly review the notion of amenability and explore its connection with bounded cohomology. We will then focus on the aforementioned result. We will begin with the group of homeomorphisms of R^n with compact support, as treated by Matsumoto and Morita in 1985, and then turn on the full groups of all the homeomorphisms and diffeomorphisms of Euclidean spaces. The latter result is recent and is due to Fournier-Facio, Monod, and Nariman (2024). Quite surprisingly, in both cases the bounded cohomology vanishes, although the techniques employed are substantially different