One, Preparatory -- 0. Algebras, Modules, Complexes -- I. Cohomology Groups and Problems Giving Rise to Them -- II. Tensor Product -- Two, Basic -- III. Homological Concepts (General Properties) -- IV. Projectivity -- V. Resolutions and Dimensions -- VI. Multi-Operational Holomorphic Calculus on the Taylor Spectrum -- VII. Flatness and Amenability -- Appendix A. Paracompact topological spaces -- Appendix B. Invariant means on locally compact groups -- Postscript -- ยง1. Extensions and derivations -- ยง2. Normal cohomology and its expression in terms of Ext -- ยง4. An interpretation of amenability-according-to-Connes in terms of the diagonal and reduced bifunctionals -- ยง5. โ{128}{156}General homologicalโ{128}{157} background to amenability according to Connes -- ยง6. Central contractibility (= central separability) and central cohomology -- ยง7. Homological dimensions. Results of a general character and results connected with the geometry of Banach spaces -- ยง8. Homological dimensions (continued). Algebras of smooth functions and some radical algebras -- ยง10. Homological dimensions (concluded). Connections with the question of an analytic structure on the spectrum -- ยง11. Miscellaneous results about the homological invariants of operator algebras and their modules -- ยง12. Completely bounded cohomology and its applications -- ยง13. Weakly amenable Banach algebras and various conditions for โ{128}{156}ordinaryโ{128}{157} and weak amenability -- ยง15. Some remarks about the development (and metamorphosis) of the problems of a multi-operator holomorphic calculus -- References -- Postscript references -- Index of terminology -- Index of notation and abbreviations