The central theme of this paper is a product formula for (continuous) bounded cohomology, and more specifically its applications to rigidity theory for lattices — both in Lie/algebraic groups and more general topological groups. A more condensed exposition of some of the material published in the Lecture Note of the second author is followed by finiteness results for lattices. An appendix by Burger–Iozzi pins down a powerful use of boundary maps in this context.