The purpose of this monograph is (a) to lay the foundations for a conceptual approach to bounded cohomology; (b) to harvest the resulting applications in rigidity theory. Of central importance is the new interplay between measure theory, amenability, Banach representations on one hand, with the homological apparatus on the other hand. The applications obtained in this text include rigidity for actions on Teichmüller spaces and homeomorphisms of the circle. The main tools include Poisson boundaries for random walks, spectral sequences, Zimmer-amenability, cohomological induction.