Bounded cohomology of groups is a variant of ordinary group cohomology introduced by Johnson in the 70s in the context of Banach algebras and then intensively studied by Gromov in his seminal paper "Volume and bounded cohomology" in relation to geometry and topology of manifolds. Since the 80s bounded cohomology has then grown up as an independent and active research field. On the other hand, it is notoriously hard to compute bounded cohomology. For this reason it is natural to first investigate groups with trivial bounded cohomology groups. During this talk we survey recent advances around "acyclicity" in bounded cohomology and we will introduce a new algebraic criterion for the vanishing of bounded cohomology. This is part of a joint work with Caterina Campagnolo, Francesco Fournier-Facio and Yash Lodha.