We say that an increasing infinite sequence of groups is (co)homologically stable if for every degree in (co)homology there is a term in the sequence after which all subsequent inclusions induce isomorphisms. In classical group homology, this is a feature exhibited by a wide range of families of groups, such as symmetric groups, mapping class groups, classical linear groups, among others. In the realm of bounded cohomology (in the sense of Gromov), where computations are scarce in the literature, stabilization results happens to be a useful computational tool. After a brief introduction to bounded cohomology, I will explain how the problem of determining the bounded cohomology of a classical simple Lie group can be approached via the bounded-cohomological stability of its corresponding family. Then, I will present a new stability theorem in bounded cohomology for families of classical simple Lie groups, and some computations. Based on joint work with Tobias Hartnick.