Continuous bounded cohomology of locally compact groups Recent research has repeatedly led to connections between important rigidity questions and bounded cohomology. However the latter has remained by and large intractable. This monograph introduces the functorial study of the continuous bounded cohomology for topological groups with coefficients in Banach modules. The powerful techniques of this more general theory have successfully solved a number of the original problems in bounded cohomology. As applications one obtains in particular rigidity results for actions on the circle for representations on complex hyperbolic spaces and on Teichmüller spaces. A special effort has been made to provide detailed proofs or references in quite some generality. Introduction; Chapter I: Banach modules $Linfty$ spaces: Banach modules $L^ infty$ spaces Integration. Chapter II: Relative injectivity and amenable actions: Relative injectivity Amenability and amenable actions. Chapter III: Definition and characterization of continuous bounded cohomology: A naive definition The functorial characterization Functoriality Continuous cohomology and the comparison map. Chapter IV: Cohomological techniques: General techniques Double ergodicity Hochschild Serre spectral Sequence. Chapter V: Towards applications: Interpretations of $( rm EH)^2 ( rm cb)$ General irreducible lattices. Bibliography. Index.