Part I. Observations of Black Holes: 1. Black holes in our Galaxy: observations P. Charles 2. Black holes in Active Galactic Nuclei: observations G. M. Madejski Part II. Physics Close to a Black Hole: 3. Physics of black holes I. D. Novikov 4. Physics of black hole accretion M. A. Abramowicz Part III. Turbulence, Viscosity: 5. Disc turbulence and viscosity A. Brandenburg Part IV. Radiative Processes: 6. The role of electron-positron pairs in accretion flows G. Bjornsson 7. Accretion disc-corona models and X/Y-ray spectra of accreting black holes J. Poutanen 8. Emission lines: signatures of relativistic rotation A. C. Fabian Part V. Accretion Discs: 9. Spectral tests of models for accretion disks around black holes J. H. Krolik 10. Advection-dominated accretion around black holes R. Narayan, R. Mahadevan and E. Quataert 11. Accretion disc instabilities and advection dominated accretion flows J.-P. Lasota 12. Magnetic field and multi-phase gas in AGN A. Celotti and M. J. Rees Part V. Discs in Binary Black Holes: 13. Supermassive binary black holes in galaxies P. Artymowicz Part VI. Stability of Accretion Discs: 14. Large scale perturbation of an accretion disc by a black hole binary companion J. C. B. Papaloizou, C. Terquem and D. N. C. Lin 15. Stable oscillations of black hole accretion discs M. Nowak and D. Lehr Part VI. Coherant Structures: 16. Spotted discs A. Bracco, A. Provenzale, E. A. Spiegel and P. Yecko Self-organized critically in accretion discs P. Wiita and Y. Xiong Summary: old and new advances in black hole accretion disc theory R. Svensson.