Promoted by the seminal work of Gromov, bounded cohomology is a powerful and by now well-established tool to study properties of groups and spaces. However, it is often difficult to compute and, beyond the case of amenable groups, many questions remain widely open. In joint work with Francesco Fournier-Facio, Yash Lodha and Marco Moraschini, we present a new algebraic condition that implies the vanishing of the bounded cohomology of a given group for a big family of coefficient modules. This condition is satisfied by many non-amenable groups of topological, geometric or algebraic origin, and even, surprisingly, by the generic countable group (in a precise sense). In the first part we will present the context and examples, and in the second part we will give some ideas beyond the proof of this result.